LECTURES ON RIEMANN SURFACES BY
R. C. GUMNNING
PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS 1966
MATHEMATICAL NOTES Edited by Wu-chung Hsiang, John Milnor, and Elias M. Stein Preliminary Informal Notes of University Courses and Seminars in Mathematics
1. Lectures on the h-Cobordism Theorem, by JosN MILNOR 2. Lectures on Riemann Surfaces, by RoBERT C. GUNNING 3. Dynamical Theories of Brownian Motion, by EDWARD NELSON
4. Homology of Cell Complexes, by GEORGE E. Coox and Ross L. FINNEY (based on lecture notes by NORMAN E. STEENROD) 5. Tensor Analysis, by EDWARD NELSON
6. Lectures on Vector Bundles Over Riemann Surfaces, by RoBERT C. GuNNwo 7. Notes on Cobordism Theory, by ROBERT E. STONG 8. Stationary Stochastic Processes, by TAUEYUKI HmA 9. Topics in Dynamics---I: Flows, by EDWARD NELSON
10. Lectures on Complex Analytic Varieties: The Local Parametrization Theorem, by ROBERT C. GUNNING
11. Boundary Behavior of Holomorphic Functions of Several Complex Variables, by E. M. STEIN
12. Lectures on Riemann Surfaces: Jacobi Varieties, by R. C. GUNNING
13. Topics in Algebraic and Analytic Geometry, by PHILLIP GRIF'Frrxs and JOHN AOAMS
14. Lectures on Complex Analytic Varieties: Finite Analytic Mappings, by it C. GuNNING
A complete catalogue of Princeton mathematics and science books, with prices, is available upon request.
PRINCETON UNIVERSITY PRESS
Princeton, New Jersey 08540
LECTURES ON RIEMANN SURFACES BY R. C. GUNNING
Preliminary Informal Notes of University Courses and Seminars in Mathematics
MATHEMATICAL NOTES PRINCETON UNIVERSITY PRESS
Copyright (D1966, by Princeton University Press All Rights Reserved
Second Printing, 1968
Printed in the United States of America
Preface.
These are notes for a course of lectures given at Princeton University during the academic year 1965-66.
The
subject of the lectures was compact Riemann surfaces, considered as complex analytic manifolds.
There are already several expo-
sitions of this subject from basically the some point of view; the foremost is undoubtedly Hermann Weyl's classic "Die Idee der Riemannschen Fllkche," and most of the later treatments have followed Weyl's approach to a large degree.
During recent years
there has been considerable activity in the study of complex analytic manifolds of several dimensions, and various new tools and approaches have been developed.
The aim of the lectures, in
addition to treating of a beautiful subject for its own sake, was to introduce the students to some of these techniques in the case of one complex variable, where things are simpler and the results more complete.
The material covered is indicated by the table of contents.
No familiarity with manifolds, sheaves, or sheaf coho-
mology was assumed, so those subjects are developed ab initio, although only so far as necessary for the purposes of the lectures. On the other hand, no attempt was made to discuss in detail the topology of surfaces; for that is really another subject, and there are very good treatments available elsewhere.
The basic
analytic tool' used was the Serre duality theorem, rather than the
theory of harmonic integrals or harmonic functions.
The detailed
treatment of the analytic properties of compact Riemann surfaces
begins only in §7.
Unfortunately, there was not enough time to
get very far in the discussion; so the lectures have the air of
being but an introduction to the subject. of the surprising omissions, also.
This may explain some
I hope to have an opportunity
to continue the discussion further sometime. With the possible exception of parts of §9, there is nothing really new here.
References to the literature are
scattered throughout, with no attempt at completeness.
In addition
to these and to the book of Hermann Weyl, the following general references should be mentioned here:
Paul Appell and Edouard
Goursat, "Theorie des Ptmctions Alge"briques," (Gauthier-Villars, 1930);
Kurt Hensel and Georg Landsberg, "Theorie der algebraischen
Funktionen einer Variablen,"
(Teubner, 1902; Chelsea, 1962); and
Jean-Pierre Serre, "Corps locaux," (Hermann, 1962). I should like to express my thanks here to Richard Hamilton, Henry Laufer, and Richard Mandelbaum for many suggestions and improvements; and to Elizabeth Epstein for typing the manuscript.
Princeton, New Jersey May, 1966
R. C. Gunning
Contents Page §1.
Fundamental definitions
.
.
.
.
.
.
.
.
.
.
.
.
.
.
complex analytic structures; manifolds; b. holomorphic functions; a. holomorphic tori. f. mappings; e. spheres; a. c.
§2.
Sheaves . a.
d.
§3.
. .
. . . . . . . . . . . . . .
sheaves; b. presheaves; sheaf homomorphisms.
Cohomology . . . .
.
. . .
.
.
14
examples;
c.
. . . . . . . . . .
27
. . . . .
.
cohomology of cohomologyr of a covering; b. exact cohomology sequence; c. a space; Dolbeault's theorem; e. d. fine sheaves; leray's theorem. f. a.
§4.
Divisors and line bundles . divisors; theorems. a.
§5.
b.
.
.
.
.
.
.
.
.
.
.
.
.
.
48
finiteness
line bundles; c.
Differential forms and Serre duality .
.
.
.
.
.
68
.
.
.
8o
. . . .
.
98
.
.
Serre's duality b. differential forms; a. c. the canonical bundle. theorem; §6.
Proof of Serre's duality theorem . a. c. d.
§7.
.
d.
.
.
.
. . . .
c.
point bundles;
d.
. .
.
.
Riemann-Roch Weierstrass
Picard and Jacobi varieties . . . . . . . a. topologically trivial line bundles; b.
§9.
.
characteristic classes; b.
theorem; points. §8.
.
distributions; b. regularity theorems; distributions on a Riemann surface; proof of the theorem.
Riemann-Roch theorem . .
a.
.
complex tori; c. Abel's theorem.
.
. .
.
.
129
Riemsxm matrices;
Uniformization . . . . . . . . . . . . . . . a. affine and projective structures and connections; b. the coordinate cohomology the underlying vector bundle; c. class; e. geometrical d. Eichler cohomology; realization.
.
. . .
164
Page §10.
Representations of Riemann surfaces a.
b. c.
Appendix:
Index .
. .
.
.
.
.
.
.
the topology of surfaces
.
.
.
.
.
.
.
.
.
.
.
.
251
. . . . . . . . . . . . . . . . . . . . . . . . 2 5c
C
complex field
C
sheaf of germs of continuous functions, 22
,q'
divisor, 1+9
Cm differential forms, 68
Go
sheaf of germs of
/L
sheaf of germs of distributions, 83
6"
sheaf of germs of holomorphic functions, 20
1,0
220
.
branched coverings of the sphere; algebraic plane curves; the principal curve.
SOME FREQUENTLY USED SYMBOIB
&'
.
sheaf of germs of Abelian differentials, 72
]P
projective line
K
canonical line bundle, 78
7(g) =
dim r(M, B' (g)), 1,11
§1.
Fundamental definitions.
(a)
The field of real numbers will be denoted by
e
and the field
of complex numbers by C ; both are topological fields, with the familiar structures.
The cartesian product of
the usual euclidean n-space, will be denoted by topological spaces, Definition.
C and
Ht with itself
n times,
fl ; note that, as
can be identified with one another.
An n-dimensional topological manifold is a
p E M has an open neighbor-
Hausdorff space M such that every point hood homeomorphic to an open cell in
fl
.
Let M be an n-dimensional topological manifold. A coordinate covering
(Ua,za}
of M consists of an open covering
{Ua}
of M
together with homeomorphisms
za: Ua -- Va from the subsets Ua C M to open cells
admits a coordinate covering.
The sets
and the mappings
be called coordinate neighborhoods, called coordinate mappings.
Va C t' .
Ua will
za will be
By definition, any topological manifold
On each non-empty intersection Ua n U
two different homeomorphisms into IF
fad = za o zl: zP(Ua n
are defined; the compositions
T za(Ua n u,)
will be called the coordinate transition functions of the coordinate covering.
Thus for a point p e Ua fl u, , the two coordinate mappings
are related by
za(p) = fa3(zP(p))
illustrate these concepts.
-1-
.
The following diagram should
Page §10.
Representations of Riemann surfaces a.
b. c.
Appendix:
.
.
.
.
.
.
.
220
.
branched coverings of the sphere; algebraic plane curves; the principal curve.
the topology of surfaces
.
.
.
.
.
.
.
.
.
.
.
.
251
255 SOME FREQUENTLY USED SYMB0L4
C
complex field
C
sheaf of germs of continuous functions, 22
t9-
divisor, 49
sheaf of germs of C- differential forms, 68 sheaf of germs of distributions, 83 sheaf of germs of holomorphic functions, 20
1,0 &1
sheaf of germs of Abelian differentials, 72
7P
projective line
K
canonical line bundle, 78
7(9) = dim r(M, er(g)), 171
§1.
Fundamental definitions.
(a)
The field of real numbers will be denoted by R , and the field
of complex numbers by C ; both are topological fields, with the familiar structures.
The cartesian product of I with itself n times,
the usual euclidean n-space, will be denoted by HP ; note that, as topological spaces, Definition.
C and l?
can be identified with one another.
An n-dimensional topological manifold is a
Hausdorff space M such that every point p e M has an open neighborhood homeomorphic to an open cell in H' .
let M be an n-dimensional topological manifold. A coordinate covering
(Ua,za}
of M consists of an open covering
(Ua)
of M
together with homeamorphisms
za: Ua --
Va
from the subsets Ua C M to open cells
admits a coordinate covering.
The sets
U. will
za will be
and the mappings
be called coordinate neighborhoods, called coordinate mappings.
Va C IF .
By definition, any topological manifold On each non-empty intersection Ua fl u,
two different homeomorphisms into
IF are defined; the compositions
fa3 = za a z1: zP(Ua n
u
will be called the coordinate transition functions of the coordinate covering.
Thus for a point p e Ua fl u, , the two coordinate mappings
are related by
za(p) = fa3 (zP(p))
illustrate these concepts.
-1-
.
The following diagram should
M
(Ua n UU
shaded)
and
z,(u(, n u,) shaded)
(Va)
and of the mappings
(z,,(U,, n u,,)
Note that a description of the sets
is enough to reconstruct the original manifold M ; for M can
(fa$)
be obtained from the disjoint union of all the sets fying a point
za a Va and a point
z
As a convenient abbreviation, the sets
(Va)
whenever
a VV
by identi-
za = fa,3(zd
V. will sometimes also be
called coordinate neighborhoods for M Suppose that
(Ua,za)
coverings of the manifold M .
and
CUOIL,za)
The union
are two coordinate (Ua,za) U (U&,za)
of these
two coordinate coverings is the new coordinate covering consisting of
all the coordinate neighborhoods and mappings from the two given coverings.
It is important to observe that the set of coordinate
transition functions for the union
(Ua,za) U (U'a,za)
is properly
larger than the union of the sets of coordinate transition-functions for
(Ua,za)
and for
(Ua,za) ; for, in addition to the coordinate
transition functions associated to the intersections those associated to the intersections
ua, n ut
Ua n U
, there are the coor-
dinate transition functions associated to all the intersections ua n u,
and
0 For most of the subsequent discussion, the manifolds under
discussion will be of dimension 2; and the coordinate neighborhoods
(Va)
will be considered as lying in the complex line
in i .
The coordinate transition functions
za = f
C rather than (zP)
are hence
continuous complex-valued functions defined on subsets of the complex line
C .
All manifolds will be assumed connected.
Coordinate coverings having particular properties can be used
(b)
to impose a great many additional structures on topological manifolds. A coordinate covering
{Ua,za)
of a 2-dimnsional manifold M will
be called a complex analytic coordinate covering if all the coordinate transition functions are holomorphic (that is.. complex analytic) functions.
Two complex analytic coordinate coverings will be called
equivalent if their union is also a complex analytic coordinate covering.
It is easy to see that this is indeed a proper equivalence
relation.
(Since symmetry and reflexivity are trivial, it is only
necessary to verify transitivity. analytic coordinate coverings
(Ua, za)
equivalent to (Y' , z")
Consider, therefore, complex
(Uaza} .
suitably small open neighborhood of
and
such that z" (P)
fyP , with the obvious notation.
fyP =
(Ua,z&) , and
For any point p e Ua fl t3
will be a coordinate neighborhood Uy
fad = fa7
equivalent to
there
p e Uy ; and in a
it is obvious that Since
faY = za a (zy)-1
are holomorphic by iypothesis, and since any
o
composition of holomorphic functions is again holomorphic, it follows
that fcep = za o (z')-' is holomorphic near all points
p eU. fl Till
transitivity.)
z" (P) .
This holds for
, and that suffices to prove the desired
An equivalence class of complex analytic coordinate
-3-
coverings of M will be called a complex analytic structure (or simply a complex structure) on m .
in the traditional terminology, a surface
M with a fixed complex structure is called a Riemann surface. It should be noted that the only property of holomorphic
functions needed for the preceding definitions is that holomorphic functions are closed under composition, whenever composition is defined; this will be called the pseudogroup property.
Thus, for any class of
homeomorphisms with the pseudogroup property, it is possible to introduce a corresponding structure on manifolds.
(For a general discussion
of pseudogroups and their classification, see for instance V. W. Guillemin and S. Sternberg, "An algebraic model of transitive differential geometry," Bull. Amer. Math. Soc. 70 (1961+), 16-47, and the
literature mentioned there.)
of EF
As an example, the set of homeamorphisms
which possess continuous partial derivatives of all orders has
the pseudogroup property; the corresponding structure on a manifold will be called a differentiable structure, (or more precisely, a differentiable structure).
CO
Since holomorphic functions are also
infinitely differentiable, a complex analytic structure on a manifold belongs to a unique differentiable structure; the complex analytic structure will be said to be subordinate to the differentiable structure.
These differentiable structures do not play a significant role
in the study of Riemann surfaces, since in fact there is a unique differentiable structure on any 2-dimensional manifold; (see J. R. Munlares, "Obstructions to the smoothing of piecewise-differentiable
homeomorphisms," Annals of Math. 72 (1960), 521-554).
_4_
This situation
is quite different in the higher-dimensional cases, however.
Some further pseudogroups and related structures, subordinate to complex analytic structures, will appear later in this discussion.
(c)
It M be a Riemaan surface, and
(Ua,za)
be a complex
analytic coordinate covering belonging to the given complex analytic
structure on M . complex line
A mapping f from an open subset U C M into the
C will be called a holomorphic function in U
if for
0 the mapping
each intArsection u fl U.
fe to: za(UflUa)--->C is a holomorphic function in the subset
za(U fl Ua) C C
.
It is easy
to see that the property that a function be holomorphic is independent of the choice of complex analytic coordinate covering belonging to the complex structure; the verification will be left to the reader.
The
set of all functions holomorphic in U will be called the ring of holomorphic functions in U , and will be denoted by 0'u ; this set is clearly a ring, under the pointwise addition and multiplication of functions, and contains the constant-valued functions as a subring isomorphic to
C .
In terms of the differentiable structure associated
to the complex analytic structure of M , a differentiable function, (or more precisely, a spondingly as a mapping in each set
C"
complex-valued function) is defined corre-
f: U T C such that for
f o za is
Co"
za(U fl Ua) # 0 ; the ring of differentiable functions in
U will be denoted by C U , and the ring of all continuous complex-
valued functions in U will be denoted by a U .
Note that these
these rings are related as follows:
CU C
&U C
C
00 C C°U
U The field of meromorphic functions in an open subset U C M can be defined correspondingly as well, and will be denoted by
"EU
It should be noted that a meromorphic function is not, properly
speaking, a mapping into C ; thus the field
is not really com-
lU parable to the rings
(o° U or
CU
An interpretation of mU as a
.
set of mappings will be given later, however; (see part (e) of §1). Let
f E
for an open set
1-0'U
The order of the function
p e U .
f
U C M , and consider a point
at the point p
f e is at the point
be the order of the holomorphic function
za(p) e C
,
is defined to
for any coordinate neighborhood Ua containing
this order will be denoted
vp(f) .
it ; and
Recall that the order of a holo-
morphic function of a complex variable
z
at a point
z = a
is the
order of the first non-zero coefficient in the Taylor expansion of the function in terms of the variable
z - a ; and note that the order is
independent of the complex analytic coordinate covering belonging to
the complex analytic structure of M . points
p e U ; and
of U , if
f
p(f) > 0
at all
only at a discrete subset of points
is not identically zero.
Similarly, for a meromorphic
f eU , the order can be defined, and will also be denoted
function
p(f) ; in this case, v (f)
vp(f) > 0
Of course,
0
vp(f)
can be negative as well, but again _
except at a discrete subset of points of U .
p Lemma 1.
If M
is a compact connected Riemann surface,
then B'M=C Proof.
As noted above, the ring of constant-valued funct ons
is a subring C C &M . the function
Ua
IfI
If
f e
B'M
, then since M
is compact,
must attain its maximum at some point p e M
is a coordinate neighborhood containing p
its maximum at an interior point
,
then
z11
if ,
za(p) a za(U n Ua) ; hence
must be constant in an open neighborhood of
constant in an open neighborhood of p .
If reaches
f
zal
f must be
za(p) , and
It follows directly from
the identity theorem in function theory that the interior of the set
of points at which f
is constant is both open and closed in M
since that set is non-empty, and M is connected,
f
is actually
constant on M and thus d M C C.
(d)
The notion of a holomorphic function on a Riemann surface can
Let M and
be generalized as follows.
and let
(Uaza}
and
(Ua,z&}
M'
be two Riemann surfaces;
be complex analytic coordinate coverings
belonging to the two given complex structures. A mapping
f: M --> M'
is called a holomorphic mapping if, for any point p e M and for any coordinate neighborhoods U. C M, f(p) a U1 , the function
U1 C M'
such that p e U. and
z' o f o za is a holomorphic function in
the usual sense in some open neighborhood of the point
za(p) e C
.
It is easy to see that the property that a mapping be holomorphic is independent of the choices of complex analytic coordinate coverings
belonging to the two complex analytic structures; the verification will be left to the reader.
Note that a holomorphic mapping is neces-
sarily continuous; morever, such a mapping is also differentiable (or more precisely,
C") in terms of the differentiable structures on the
surfaces. A holomorphie function is the special case of a holomorphie
mapping from M to the Riemam surface
C .
These holomorphie mappings can be characterized very conveniently by their effects on holomorphic functions.
be any continuous mapping between two Riemann surfaces U' C M' f*:
be any open subset of M'
(U' -- 4U
, where
M,M' , and let
induces a homomorphism
U = f-1(U') C M, by defining
f*( dqUt) C CU is a well defined
f ; in particular,
f*(hU,) = hU,
f
The map
.
f: M -4 M'
Let
subring.
A continuous mapping
Lemma 2.
between two
f: M --? M'
*
Riemann surfaces is a holomorphic mappinggif and only if f ( OU' ) C OU for every open subset U' C M' , where U = f-1 (U') . Proof. Ua C M, f
Select any point
p e M , and coordinate neighborhoods
such that p e Ua and
U'' C M'
f(p) a U'
.
If the mapping
satisfies the conditions of the lemma, then considering in par-
ticular the holomorphic function f*(zl) = z' o f
f
f
za1
oU' , it follows that z
is a holomorphic mapping.
is holomorphic and if
an open neighborhood h' o f o
a
is holomorphic, that is, that
morphic; therefore
mapping
z,,
U' C U'
= (h' a
za is holo-
Conversely, if the
is a holomorphic function in
f(p) , then
of
o (z' o f
function in a neighborhood of the holomorphic functions
h'
o f ,
z-1)
f*(hl)
o
za
=
is a holomorphic
za(p) , since it is the composition of
h' o (zO)
and
z' 0 f .
z-1
; hence
f
satisfies the conditions of the lemma. Extending the previous discussion, a topological homeomorphism f: M --? M'
between two Riemann surfaces will be called a
a
olomorphic
isomorphism if both mappings
f
the Riemann surfaces M and
M'
and f -l
are holomorphic mappings;
will be called isomorphic if there
is a holomorphic isomorphism between them.
Clearly, the real interest
lies in the isomorphism classes of Riemann surfaces.
(e) C ;
The simplest example of a Riemann surface is the complex line of course, since any subdomain of a Riemann surface is again a
Riemann.surface, subdomains of C are also contenders for the title of simplest Riemann surface.
As for compact manifolds, the 2-sphere
is clearly the simplest case; thus there arises the question whether the 2-sphere admits a complex structure.
The 2-sphere, considered merely as a topological manifold M can be given a coordinate covering as follows.
Let
a, a e M be two
distinct points of M , which can be envisaged as the north and south poles of the 2-sphere.
The open sets
UO = M-s
and Ul = M-n
M , and are topological cells; so select some homeomorphisms
cover
za
between these sets and the standard 2-cell, which can be taken to be
the full euclidean plane C . (U0,z0), (U1,z1)
further that
of M .
This describes a coordinate covering
There is no loss of generality in supposing
z0(n) = 0 e C , and
transition function
f01
z1(s) = 0 e C ; so the coordinate
is a homeomorphism
f01: (C-0) -->
(C
,
which takes the interior of an open topological disc about the origin
in C -0 onto the exterior of another such set. any choice of such a homeomorphism f01
-q-
Conversely of course,
can be realized as the coor-
dinate transition function for a coordinate covering of the 2-sphere
M of the above form; the question of the existence of a complex analytic structure on the 2-sphere thus becomes merely the question
of the existence of a complex analytic homeomorphism z0 = f01(zl) this form.
In particular, the function
of
zo = f01(zl) = 1/z1 will
serve the purpose; and the 2-sphere with this complex structure will
be called the complex projective line, and denoted by R'. It should be noted that the complex line
IF
as described
above is actually the one-dimensional complex projective space in the usual sense; (see for instance W. V. D. Bodge and D. Pedoe, Methods of Algebraic Geometry, volume I, chapter V, (Cambridge University Press, 1953)).
Thus let C = C - 0 , considered as a multiplicative group;
and let C
*
act as a transformation group on the space
{(t1,t2) # (0, 0)}
quotient space
(C
by t-(t'1 1) = (tto,tt1) - 0)1C'
for
t e r.*
C 2_ .
0 = The
is the one-dimensional complex projective
space; each point of the projective space can be represented by a pair of complex numbers
(t1,t2) / (0,0)
called the homogeneous coordinates
of the point, this representation of course being far from unique.
The two sets
U0 = {Q0,y a C2it0
0}/C,
Ul = C(50, 1) a L%21 51
01
cover the projective space, and each ca41 be mapped in a one-one manner
onto the complex line
C by a mapping
z0 = Y 0 Then
(U0,z0)
and
{U1,z1}
or
z1 =
form a coordinate covering of the space,
exhibiting it as the Riemann surface P descried above.
One trivial property of the complex projective line deserves note here.
Consider an arbitrary Riemann surface M , and a holo-
morphic mapping
f: M - P .
point p e M the mapping f geneous coordinates on P by
In an open neighborhood U of each can be described in terms of the homo-
are holomorphic functions on U in the usual sense. f0(p)/f1(p)
fa(p)
f(p) - (f0(P),fl(P)) , where
Then the quotient
is a meromorphic function in U ; this function is clearly
independent of the choice of homogeneous representation, and so is
defined throughout the Riemann surface M .
Conversely, any mero-
morphic function on M can be represented locally as the quotient of holomorphic functions,
f0(p)/fl(p) ; and then
f(p) = (f0(p),fl(p))
f: M --a P .
is a well-defined holomorphic mapping
That is to say,
the meromorphic functions on M are in natural one-to-one correspondence with the holomorphic mappings
(f)
f: M ---> P P.
The next simplest compact 2-manifold is a surface of genus 1,
a torus; complex analytic structures are also easy to describe in this case.
in the complex line
C select any two complex numbers
which are linearly independent over the reels; so
complex numbers, and wl/w2 J R .
The numbers
wi
wl,w2
wl,w2
are non-zero
generate a
subgroup A C C , namely A = (nlwl + n2w2lnpn2 e Z = additive group of integers). The quotient group face of genus 1'.
C/A
is a well-defined topological space, a sur-
It is evident that
C/A has a natural structure
inherited from that of C ; as coordinate neighborhoods in
-11
C/A
take
open subsets of C which contain no points congruent to one another
modulo A .
-
In discussing the sphere, a single complex analytic structure was described; indeed, it will later be shown that there is a unique complex structure on the sphere.
In discussing the torus, there were
two arbitrary parameters Involved in the description of the complex analytic structure, the two constants
W1,W2 .
It is natural to ask
whether there are different complex analytic structures on the torus, corresponding to various choices of the parameters
W1,w2 ; this is
indeed so, showing thusly that a given topological surface may carry a variety of inequivalent complex analytic structures. that
A = (nlwl + n2w2Ini a Z)
and
Suppose then
At = Cnlwt + n2w'Ini e 2 }
are
two lattice subgroups of C , with associated Riemann surfaces
M = C/A
and M' = C/A'
.
If the surfaces M and M'
there is a topological homeomorphism f-1
are holomorphic mappings.
f
f
and
lifts to a mapping
C of M = CIA to the uni-
F from the universal covering surface versal covering surface
such that
f: M --> Mt
Now the mapping
are isomorphic,
C of M' _ C /Al ; and in view of the defi-
nition of the complex structures on M and M' , the mapping
F: C -* C must be a complex analytic mapping. Moreover, since arises from the homeomorphism
5
(1)
f: C/A ---* C,/A' , it follows that
F(z+w1) =F(z)
F(z+w2) =F(z)+a'211+e'22w2'
for some integers
aii e Z , such that
Differentiating the'above equations, Ft (z) ; so
F
F' (z)
a'11a22
a12a21 = ± 1 °
F*(z+wl) = F'(z)
and
F'(z+w2) _
is invariant under A , b ace determines a holomorphic
function on M = C/A .
Since
C/A
is compact, it follows from
Ira 1 that the function
F' (z)
Ft(z) __ c ; and therefore
F(z) = cz + d
must be constant on M , hence for some constants
c,d
There is of course no loss of generality in translating the surfaces d - 0 ; so
so that
F(z) __ cz , for some complex constant
c
0
Now from equations (1) it follows that
cwl - allw1 + a]2w2,
(2)
The complex numbers
w = wl/w2
and
cw2 = e.lw1 + a22w2 w' =
complex analytic structures of M and
associated to the are therefore related by
Mt
allwt + a12
w=
(3)
,
s'21 such that
for some integers
alla22 - a72a21 = + 1 .
Conversely,
ai3
if
w
and
w'
are so related, there is a complex constant
such that (2) holds, and the function
and therefore M and M'
are isomorphic.
surfaces M = CIA and M' = C/A' w = wl/w2
and
wt = w1t/w2
F(z) - cz
c # 0
then satisfies (1);
Consequently, the Riemann
are isomorphic if and only if
are related as above.
The set of all
possible complex structures of the above form in a complex torus thus correspond to all non-real complex numbers
w modulo the equivalence
relation (3); for a. more detailed description of the latter relation,
see for instance J. Lehner, Discontinuous Groups and Automorphic Functions, Chapter XI, (American Mathematical Society, 1964).
§2.
Sheaves
(a)
Sheaves have proved to be a very useful tool in the theory of
functions of several complex variables, and have occasionally been used in one complex variable as well.
For the purposes of an eventual
simplicity and of convenience of generalization, they will be used systematically throughout the present discussion of Riemann surfaces. However, no previous acquaintance with sheaves will be assumed here. Those readers already familiar with the general properties of sheaves,
and with the cohomology theory of sheaves, can readily skipthis and the following section. Definition., l)
A sheaf (of abelian groups) over a topological
space M is a topological space 7r:
.1 -a M (i)
(ii)
7r
,
I
, together with a mapping
such that:
is a local homeomorphism;
for each point
p e M the set
IP
= 7r
1(P) C J has the
structure of an abelian group; y
(iii)
I.
the group operations are'continuous in the topology of d
The third condition in the above definition is, more explicitly,
the following.
In the cartesian product
,6 X J with the product
topology introduce the subset
{(s1,s2) E JXJ hr(s
7r(s2)} ,
Throughout this and the following section, the discussion will be limited to sheaves of abelian groups; it is left to the reader to note the obvious modifications necessary to the consideration of sheaves of rings or fields, etc.
-14+
with the topology it inherits from the imbedding ,
.J a J - ,st
The mapping
given by
C
o
x B
(sips 2) e J.) -+ si s2 a ,aQ
is well-defined; the condition is that it be continuous. In the sheaf the mapping = 7r 1(p)
7r: J -+ M is called the projection; and the-set
is called the stalk over p .
Each stalk is an abelian
group, although different stalks may be quite different groups.
As a simple example, let
G be any abelian group with the
discrete topology; let ,1 be the space J= G X M with the product topology, and Then
J
is a sheaf over M , called a constant sheaf. Let
of M .
7r: G X M -;;b M be the natural projection mapping:
7r: J T M be a sheaf, and U C M be an open subset
A section of the sheaf J over U
f: U --->J
is a continuous mapping
such that 7r o f: U ---> U is the identity mapping; note
that necessarily
f(p) a
for any p e U .
J P = 7r 1(p)
The set of
all sections of 1 over U will be denoted by r(U,J ) point
s e J there must be an open neighborhood V of
.
For any
s
in I
such that
HIV: V --+ U is a homeomorphism between V and an open subset U C M ; the inverse map
1 (w-1 V)
: U --> V
is also a homeomorphism, hence is indeed a
section of d over U .
Therefore each point
s e pQ
is contained
in the image of some section; and the images of all such sections form a basis for the open neighborhoods of this, if 'then
f,g a r(U, J ) and if
f(p) = g(p)
for all points
s
.
f(p0) = g(po)
p
As a consequence of
for some point p
of some open set
U'
with
0
e U
PO a U' C U .
Now, if
f,g a r(U, J )
again, the mapping
f X g: p c U --s; (f(p),g(p)) e 4 x Al is a continuous mapping from U into the subset .1 o J C ,e xJ the composition of
f X g with the natural mapping APo) -- J ,
namely the mapping
f - g: p e U -* f(P) - g(p) a is therefore also a section.
That is to say, the set
group, under the pointwise addition of sections.
the map p e U --a p e j group
,
,dt
where
0p
P(U,
)
is a
The zero section,
is the zero element of the
P
is clear]y a section.
In general, it is rather difficult
P
to determine whether there are any non-trivial sections, that is, any sections other than the zero section.
In a sense, the sections of a sheaf determine the sheaf com-
(b)
pletely.
This observation can be made more precise in the following
manner.
Definition. A
rep
sheaf (of abelian groups) over a topological
space M consists of: (i)
(ii)
a basis
for the open sets of the topology of M ;
(Ua}
a separate abelian group ,fit a associated to each open set
Ua of the basis; (iii)
a homomorphism
pad: .jo --
a associated to each in-
clusion relation Ua C UP , such that
pa4,pm = pay whenever
UaCUPCUy .y To each sheaf
J
over M and basis
_1 h_
(Ua}
for the open sets
of the topology for M there is a naturally associated presheaf, which will be called the presheaf of sections of the sheaf I ; this
is the presheaf which assigns to the set Ua the abelian group ,e a = r(Ua,,1 ) , and assigns to the inclusion Ua C U0 the restriction
subset Ua .
of sections over U0 to the
r(Uu,I )
paa: r(UW J )
mapping
Conversely, to any presheaf
(Ua, ,1a, pa3)
over M
there is an associated sheaf, which is constructed as follows.
For
each point p e M consider the collection ZL (p) = (UaIp a Ua) ; this"set is partially ordered under inclusion.
p = U
u n i o n J
f0 e J
write
,
a ; and. for any two elements
e U
O:
Form the disjoint
fa " f0
if there exists a set U
fa E 1a ,
e ZC(p)
p
such
7
that Uy C Ua fl U0
and
pyafa = p70f0 .
It is a straightforward
exercise to verify that this is an equivalence relation.
equivalence classes in J p will be denoted by J
.
The set of
For any set Ua
P there is a natural mapping element
f
a
e
ppa: ;a - +
which assigns to an
Ja its equivalence class in
Again it is a simple p
matter to verify that these mappings
ppa induce on the set J p
structure of an abelian group, in such a manner that the mappings are group homomorphisms.
{J a}
(The group J p
the
ppa
constructed from the family
as above is called the direct limit group,
d p = dir. lim. U
a e u
(p) Ja ; for a more general discussion of this
concept, see S. Eilenberg and N. E. Steenrod, Foundations of Algebraic Topology, Chapter VIII, (Princeton University Press, 1952)). space of the sheaf is defined to be the set
The
Up
with the projection mapping
E M JP , --> M given by
7r:
7r( J p) = p
.
As
a basis for the open sets to define the topology of J take sets of the form
Ppa(fa) C J
[ fa] = Up E Ua for the various elements
fa e ) a .
(To see that these sets do form
the basis for a topology, it is necessary to show that for any point
s e [f] n [fP] C J there is an element
f
E J y
so that
7
Now if
s e [fy] C [fa] n [fPI
then p e Ua n UP
and ppa = ppPfP ; by definition of the mapping
Ppa , there mist be a set pyafa = pypf,3 .
U7
Therefore
such that p e U7 C Ua fl UP and
s e [fy] C [fa] n [f] , as desired.)
this topology, it is clear that the mapping homeomorphism.
and if p = 7r(s)
s e [fa] n
7r: 4 T M is a local
Finally, to show that J is a sheaf, it merely remains
to verify that the group operations are continuous.
(s1,s2) a J o J
and let f2P2
and any open neighborhood
such that
[fa]
Select any point about
sl- s2
Further, select elements f1
p = 7r(s1) = 7r(s2) .
a ,e
ppp1(fly1)
and
sl
ppP2(f2'32)
a
,dp
s2
and Then
2
ppa(fa)
ppP2(f202) ; so by the definition of the mapping
ppp1(f1P1) =
-
Ppa , there must be a set U
such that
7
Pya(fa) = PyP(fly ) -P7P(f2p ) 2 1 2 1
Now under the mapping J of --> J it follows that ([Pyf (fly )] x [pyP2 (f2,2)])n J.1 1 1 (sl,s2)
With
which maps into
is an open neighborhood of
[fa] , proving the continuity.
-18-
starting with a sheaf J , form the presheaf of sections of j {Ua)
for some basis
for the topology of M .
It is clear from the pre-
ceding construction that the associated sheaf of the presheaf of sections
of A is canonically isomorphic to I itself.
In this order, the two
constructions introduced above are thus inverse to one another.
It is
not true, however, that these constructions are inverse to one another in the other order; that is, the sheaf of sections of the associated
sheaf to a given presheaf is not always isomorphic to the given presheaf. For example, a presheaf in which
a
= Z
for all a , and
is
pad
the zero homomorphism, has the zero sheaf as its associated sheaf; and the presheaf of sections of the zero sheaf associates the zero group
to each Ua .
Clearly the problem is to characterize those presheaves
which arise as the presheaf of sections of some sheaf.
Definition. A presheaf
(Ua,
.Jc , pad)
over a topological
space M is called a complete presheaf if, whenever of the basis
a subcollection U0, {U
U0 = UP UP
for
{Ua} , the following two
conditions are fulfilled: (i)
if
fo,g0 e j o
then (ii)
if
fo
are such that
pPo fo = p0ogo
for all Up ,
= go ;
fP E J
are elements such that P
pyO fP
1 1
whenever Uy C U0
fl UP
1
2
there is a n element fo e
for any elements
j0
= py0fP 2
2
Uy of the basis,
such that fP = p fo for
all UP . Lemma 3.
A presheaf
(Ua, J a, pa$)
over a topological space
M is the presheaf of sections of some sheaf over M if and only if it is complete.
Proof.
It is obvious that the presheaf of sections of any
sheaf is complete; the converse assertion is the one of interest.
J be the associated. sheaf to the given presheaf.
There is a canonical
P: ja --> r(Ua, J ) defined by p(fa) _ [fa] =
homomorphism U
Let
E U ppa(fa)
a
It suffices to prove that
for each set Ua .
Suppose firstly that
Ppa(Fa) = 0
for all p e Ua .
exist a set
UP
is an isomorphism,
P(fa) = 0 , that is, that
Then for each point
such thatN p E Up C Ua and
these sets (Ucover Ua , presheaf that
p
.
p e Ua there must
poa(fa)
Since
it follows from property (i) of a complete
Next, consider any section
fa = 0 .
0 .
f e r(Ua, al
)
.
For each point
p e Ua there must exist a set UP with p e UP C Ua ,
and an element
f13 e 1j
such that
ppp(fP) = f(p)
.
P
and
[fe]
coincide at
so by restricting UP points
p , hence in a full open neighborhood of p ; further if necessary,
The seta (UJ)
q e Up .
The sections f
pgP(fP) = f(q)
for all
cover Ua , and obviously satisfy
condition (ii) of the definition of a complete presheaf; therefore
there is some element that
fa e
Let M be a Riemenn surface, and
open sets in the topology of M .
Ua
P
Cfa) = ff , hence such
This suffices to complete the proof.
f = [fa}
(c)
(9-
such that
(Ua)
be any basis for the
To each set Ua associate the ring
of functions holomorphic in Ua ; and to each inclusion relation
Ua C UP associate the natural restriction mapping Pte: 5U --> NU Clearly
(Ua, &
pad)
a 0 is a presheaf over M ; the associated sheaf
U al
is called the sheaf of germs of holomorphie functions on M , and will
be denoted by 0
.
(For the purposes at hand, it is 'the additive
structure of the rings c sheaf of abelian groups.
which will be considered, to obtain a
Ua
Actually, of course,
(.
is a sheaf of rings
over the space M , using the obvious modifications of the preceding definitions.
The ring structure plays a very important role in the
case of holomorphic functions of several complex variables; for more in this direction, see R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Chapter IV, (Prentice-Hall, 1965).)
To interpret the stalk Q-p of the sheaf
OL at a point
p c M, select a coordinate neighborhood U containing p coordinate mapping
z: U --> V C C such that
struction of the stalk 61 p
z(p) = 0
0
The con-
.
being local, it is sufficient to consider
the entire construction within the set V C C . borhood of
and a
To each open neigh-
in V consider the ring of functions holomorphic in
that neighborhood.
Two such functions are equivalent if they agree
in some smaller open neighborhood of D ; and the equivalence classes, called the germs of holomorphic functions at
0 , form the stalk 61p .
To each function holomorphic in an open neighborhood of its power series expansion at the point
0
associate
0 ; equivalent functions
clearly determine the same power series, and every convergent power series arises from some unique germ.
the ring C (z)
Therefore
61 p
is isomorphic to
of convergent, complex power series in the variable z.
Note that the presheaf
(U., &" , pa$)
is obviously complete;
a hence with the natural isomorphism it is possible to identify
r(Ua, () = 8U
.
That is, the sections of the sheaf d over any
a open set U C M are identified with the ring 6 U of functions holomorphic in U .
A similar construction, beginning with the multiplicative
0U
groups
to the sheaf
of nowhere-vanishing holomorphic functions on U leads Q-
of germs of nowhere-vanishing holomorphic functions;
again, in the same manner, it is possible to introduce the sheaf
germs of meromorphic functions on M ,
of
(a sheaf of fields, actually),
of germs of meromorphic functions
or the multiplicative sheaf
not identically zero on M . Considering merely the differentiable structure of M , and the rings
an U or G U of infinitely differentiable or continuous
functions on open sets U C M , leads in a similar manner to the sheaf
a 00 of germs of
C°°
functions on m , or the sheaf !o of germs of
continuous functions on M .
Note that there are the natural inclusion
relations
6C dOC C Various-r-elations between sheaves over a fixed topological
(d)
space M are of importance in the applications. I
First, for a sheaf
over M , let E C M be an arbitrary subset of M .
restriction of the sheaf where
.d
to
E
is the subset
The
fir 1(E) C I ,
7r: J - M is the projection mapping; the restriction, which
will be denoted by .J IE , is clearly a sheaf over the set particular, for a point
J over p .
p e M ,
d lp =
dp
E .
In
is just the stalk of
(For example, if M is a Riemann surface, then an
open subset U C M is also a Riemann surface, and its sheaf of germs of holomorphic functions is just 6JU , the restriction of the sheaf (9- over
the sheaf
M to the subset U . 19 1 E
If E C M is not an open subset,
cannot generally be interpreted as a subsheaf of the
sheaf of germs of continuous functions on the space
case in which E
E
itself; the
is a single point of M illustrates why.)
Again let J be a sheaf of abelian groups over M , and let
P, CJ be a subset of i (i)
(ii)
.
Then
.
is an open subset of
.
is called a subsheaf of ,g
if:
; and
for each point p e M, R p = X n J p
is a subgroup of 1 .
Clearly . is itself a sheaf of abelian groups over M , its projection mapping being the restriction to
The quotient sheaf 9 = ,! / P,
point p e M let and let 0 = Up e
0p =
T, of the projection mapping is then defined as follows. P, p
,Qp/
given by 7r(g p ) = p . The natural mapping in J and in
w:
'? --;- M
which
,a{
q):
associates to arty element of 4 p its coset in it
For each
be the natural quotient group;
M ap , with the projection mapping
mutes with the projections
of J
7r
,JP/
p=
com-
gc,p
Introduce on 0 the
.
natural quotient topology, defining a set U C a to be open if and
only if c l(U)
is open in ) .
It is an easy matter, which will be
left to the reader, to verify that 0 is then a sheaf.
For example let M be a Riemann surface, and of germs of holomorphic functions on M , as usual.
0- be the sheaf Let
P = (pl,...,pn)
be a finite number of distinct points on M , and for each open subset
U C M consider the set
W U = (f e & Ulf(pi) = 0 whenever
Note that each W U
is a subgroup of 6
u
;
pi e U, i = l,...,n}
and that the groups
(2;U}
for all the open subsets of M , with the natural restriction mappings,
form a presheaf over M .
The associated sheaf is then a subsheaf
C (.
, and this leads to a quotient sheaf 3 = (Q I P .
To describe
this quotient sheaf, note that for a point p / P necessarily
PIP= QI P , hence a P = 0 .
However for a point p1 E P , R
P1
C£
P1
is the subgroup consisting of those germs of holomorphic functions
which vanish at
pl , or equivalently, the subgroup of those convergent
power series with zero constant term; thus )
;; C , = (12 / Pi pl 1 the latter ispmorphism being that which associates to any power series
in 0p
its constant term. That is to say, the sheaf ) will have 1 a trivial stalk (consisting of the zero group alone) at all points p / P , and it will have stalk 0
= C at the points
A
p e P .
P
sheaf of this sort is sometimes called a skyscraper sheaf.
Now suppose that 2 and D
are two sheaves of abelian groups
o: J -- M and
over M , with projection mappings A mapping
is called a sheaf mapping if:
cp:
tinuous; and
(ii)
r o cp = a .
for any point p e M, stalks.
cp( J
Further, for any
P
M
r:
(i)
cp
is con-
The second condition implies that,
) C D
;
P
f e I'(U,.J)
so a sheaf mapping preserves
for an open subset U C M
W , f will be a continuous mapping from U into
0 such that
-r o (c o f) = a o f = the identity; that is, q) o f e r(U,' Therefore the sheaf mapping qi :
r(U, .4 ) --
.
r(u, 'J
yields an induced mapping
qp
) .
In particular, since (f(U)) , for all
open subsets U C M and all sections
f e r(U,,4 )
,
is a basis for
the topology of J , the sheaf mapping is open as well as continuous; and since the mappings
a
and
r
are local homeomorphisms, so is
(p.
That is to say, any sheaf mapping is necessarily a local homeomorphism
between the spaces I and )
.
The sheaf mapping p is called a
sheaf homomorphism if it is a homomorphism on each stalk; the induced
mapping
is then a homomorphism between the groups of sections,
'p
called the induced homomorphism. A sheaf isomorphism is a sheaf homomorphism with an inverse which is also a sheaf homomorphism; the
notation j _ 0
will be used to indicate that the sheaves
J
and
over M are isomorphic. For example, considering again the sheaf
fP e ( associate the germ
surface M , to each germ e(fp) = exp.(2Tii p) a
e: (9--->
m.
(-over a Riemann
p
This determines a sheaf homomorphism
.
Similarly of course, considering merely the sheaves
-
of germs of continuous rather than holomorphic functions, there is a
sheaf homomorphism
e:
C
.
For any sheaf homomorphism p: J -> 3 over a space M , the kernel of
cp
is the subset of j consisting of those points which
map into the zero element of any group the subset
(p-1 (0) Cj where
0 e r(M,
p ; that is, the kernel is
is the zero section of
Since the zero section is an open subset of . , the kernel is clearly
a subsheaf of d .
The image of p
and it follows readily that isomorphism of sheaves.
is apsubsheaf of 3 as well; denotes
image(p) = ad /kernel(cp) , where
Given sheaf homomorphisms
cp:
'k---> J and
4r: J --> 0 , the diagram
will be called an exact sequence of sheaves if the image of
precisely the kernel of ' .
'p
is
Similarly, a longer string of sheaves
and sheaf homomorphisms will be called an exact sequence if for any
-25-
two consecutive homomorphisms, the image of the one is precisely the kernel of the other.
In particular, if
0
denotes the trivial sheaf
with stalk the zero group at each point of M , a sequence
is exact if and only if to a subshea iof J ),
'p
*
is an injection, (an isomorphism from
is a projection, (a homamorphism with image
all of 3 ), and the kernel of
J _ J /cp( 7, )
precisely the image of '' ; hence,
Conversely, if X is a subsheaf of 2 the
.
inclusion mapping
natural mapping
yr
is & -> J is a sheaf homomorphism; and the
tp: J --* J/ R.
is a sheaf homomorphism, such that
is an exact sequence of sheaves.
For example, on a Riemann surface M the subset Z C 6 of germs of holomorphic functions which take only integer values is a
subsheaf of & isomorphic to the constant sheaf; and this is precisely
the kernel of the sheaf homomo rp hism This homomorphism
e
e:
LQ -- Q1
is a projection, since any germ
a holomorphic logarithm near p .
introduced above. f
p
has
e
p
Therefore there arises the exact
sequence of sheaves
0 ---> Z ---* & -- &* --> 0 . Similarly, considering the sheaves of germs of continuous functions,
there is the exact sequence
0
Z -> C- 6*-->o.
Cohomolo9Y
§3.
It M be a topological space, and
(a)
covering of M .
= (Ua)
be an open
To this covering of M there is associated a sim-
plicial complex N(2 ) , called the nerve of the covering 11L , and defined as follows.
are the sets
The vertices of N(om)
Vertices UO...,Uq span a q-simplex
covering.
Ua of the
a - (U0,...,Uq)
and only if UO n ... n Uq # 0 ; the set UO n ... n Uq = (al the support of the simplex
over M . function
a
.
is called
Let ,S be a sheaf of abelian groups is a
with coefficients in the sheaf 2
A q-cochain of V
if
f which associates to every q-simplex
a e N(Vt)
a section
f(a) c r(lal,2 ) ; the set of all such q-cochains will be denoted by
Cq01L, J )
.
Whenever
f, g e C"(?JZ J )
,
their sum
f +g a Cq(%, ,J) , where (f+g)(a) = f(a) + g(o) ;
so
Cq(?A
J
,
is an abelian group. There is also an operator b: Cq(Vi , ) ) ---> Cq+1( 211 , J ) defined as follows.
If
called the coboundary operator,
f e Cq(v1 , 2 )
and
a = (UO,...,Uq+l) e N(72
then q+l
i
(Sf)(UO,...,Uq+l) = E (-1)
(1)
plalf(UO,...sUi-1'Ui+1'...aVq+l)
,
i
where
plal
denotes the restriction of the section
f(UO,...,Ui-1'Ui+1'...,Uq+l) e r(UO n ... n to
Ui-1 n Ui+1 n ...
It is clear that
lal = UO n ... n Uq+1 .
S
is a group homomorphism;
and it is a straightforward calculation to show that
Zq(Vt, J) _ (f a Cq(
subset Cq(7R
,
J )
,J )lSf = 0)
n Uq+1)
SS = 0 .
The
is a subgroup of
called the group of q-cocycles; the image
SCq-'(?A ,I ) C Cq(?1Z , j
)
is called the group of q-coboundaries,
and is a subgroup of the group of cocycles since
SB = 0 .
The quotient
group
Zq(1q ,
.
)/scq-1(Vl
j)
,
for q > 0
for q = 0
ZO(TR, .1 )
is called`lhe q-th cohomology group of 1/L. with coefficients in the
sheaf 2
. HO(b7
Lemma 4. Proof.
A zero-cochain
,
J )
r(M,I )
By the above definition,
f e C0(Zk ,2 )
set U e Vt a section
.
HO(V
,
J) = ZO( u
,
1 )
.
is a function which assigns to each
f(U) a r(U 8 ) ; and its coboundary
is a function which assigns to each pair of inter-
Sf a Cl(Vt , .1 )
setting sets U0,U1 e
a section
(5f)(UO,U1) = PUO n Ulf(U1) - PUO n Ulf(Ul) e r(UO n U1,. ) Sf = 0 , the sections f(Ua)
.
if
agree in each non-empty intersection
UO n U1 , hence altogether determine a section of. ,B
defined over the
entire space M ; and conversely, the zero-cochain defined by
restricting a global section of j over M to the various subsets
Ua
is a zero-cocycle.
(b)
This suffices to prove the desired result.
In order to have a cohomology theory associated intrinsically
to the space M , it is necessary to consider various possible
coverings of M . the covering
A covering u
= CU(X)
Va C AVa for each Va c mapping.
The covering
= (Va)
is called a refinement of
if there is a mapping ; the mapping
µ
.:
Y --a W& such that
is called a refining
may of course be a refinement of X by
various different refining mappings.
Notice that the refining mapping
induces a mapping
---> Cq ('lr , I ) ,
µ: Cq (111
If
as follows.
f e Cq( Ul , ,B
)
and o = (V0, ..., Vq) a N(`j('
, then
)
(µf)(VO,...,Vq) = P(1f(RVO,...,MVq) ; since
is a
µV0 fl :.. fl µVq J VO fl ... (1 Vq # $ , then (µV0, ...,µVq) q-simplex of N(l1 clear that
)
, and the mapping is thus well-defined.
µB = Sµ ; therefore
is a group homomorphism, and that
g
It is
determines a homomorphism
µ
µ*: Hq(A , J ) *> Hq( H
v:
u' --> n are two refining mappings, then
When
Proof.
and
) .
If If is a refinement of R , and if
Lemma 5. and
,j
v*: HO(IR ,J
q = 0 , the mappings
) T HO(' , J )
µ:
µ* = V*
µ*: HO(7? i ) - >
as follows. If f e Cq( jJ( , define
J) and
(ef)(VO,...,Vq_1)
HO(T -
are both the identity mapping, in
view of Lemma 4; so it is only necessary to consider the case In this case, construct an associated
-- ZZ
e: Cq(U1
,
q > 0
J ) T Cq-1 (1C ,.2 )
o = (V0, ...,Vq_1) e N(11
(-1)j Plalf(µVO,...,µVj,VVj,...,VVq-1)
qz J=O
o = (VO,...,Vq) a N(X
Now whenever
observe that
(Sef)(VO,...,Vq)
q+1 J-1 E E (-1)
P
i
(-1)
i=0
J=0 q
+
E
(-1)i+1 Pf(µV0,...,µVj,WVj,...,VVi-1,VVi+1,...,VVq)
i=,j+l (-1)i+1
vf(o) - µf(a)
z
J=0 -29-
Therefore, if f e Zq(7 , 2 Vf - µf = 59f , that is, that urology class.
Therefore
)
so that
Sf = 0 , it follows that
Vf and µf determine the same coho-
µ* Q V'
, as desired.
Now for any two coverings u , 14 of M , write 2r < 11Z if ?1` is a refinement of l ; the set of all coverings is partially ordered under this relation; and by Lemma 5 there is a unique homomorphism
Hq( ZQ
,
I ) -> Hq( )t ,J )
whenever if < IQ . It is clear that
these homomorphisms are transitive; hence it is possible to introduce
the direct limit group
Hq(M, ) ) = dir. lim.
Hq( V, ,
J)
,
which will be called the q-th cohomology group of the space M
with coeffie,ents in the sheaf J .
(Recall that to define the direct
limit, introduce the disjoint union
U
mology classes
Hq(
,
J. )
; and for coho-
A "j f e Hq(1R ), g e Hq( i' ,') , write
there is a refinement kP < vI and 2 < )f
, such that
f
f , g if and
g
have the same image under the natural homomorphisms
Hq( Y1
, .J
)
Hq(W, ,J
)
and Hq(y ,
j ) --> Hq(
This relation is an equivalence relation, and the set of equivalence
classes in the direct limit group
Hq(M, i ).)
there is the natural homomorphism Hq(yj
For each covering YL ,
Hq(M,.1)
.
It
follows immediately from Lemma 4 that
(2)
H0(M, A ) ° r(M, .8 ) It should be noted that for a constant sheaf such asv Z ,
the cohomology introduced above coincides with the ordinary tech
cohomology with coefficients in the group Z ; (see S. Eilenberg and N. E. Steenrod, Foundations of Algebraic Topology, Chapter ]X,
(Princeton University Press, 1952).)
(c)
Consider an exact sequence of sheaves of the following form,
over the space M :
J
0 ---.> P_
0 ---ate 0.
For any open subset U C M , the sheaf hom morphisms
cp,
induce
c*,** between the corresponding groups of sections,
homomorphisms
and there results an exact sequence of groups and homomorphisms of the form
o-->r(U,
-r(U, J)- >r(U,0)
Exactness this far is obvious, since injection of 3L
q)
.
can be considered as an
as a subsheaf of J , and *
as the passage to the
quotient sheaf; but in general the mapping r* will not have all of r(u, 3
)
as its image.
1 < Jzl < 2
in C , and consider over M the exact sequence
0 --4 Z --> & z e r(M,
any
(For instance, let M be the annulus
D*)
S * --4 0
introduced in §2(d); the function
cannot be written in the form z = exp. tai f(z)
f(z) a r(M, 1) , since necessarily
branch of log z
f(z) = tai log z
for
and no
is a single-valued holomorphic function on M .)
The cohomology theory considered above furnishes a convenient measure of the extent of the inexactness of the sequence of sections.
How-
ever, it is necessary to have further restrictions on the underlying
topological spare M . (Recall that an open covering
space M
= (Ua)
is called locally finite if each point
of a topological
p e M has an open
neighborhood V which meets at most finitely many of the sets U. .
A Hausdorff space M
is called paracompact if every open covering has
a locally finite refinement.
Any separable manifold is paracompact.
In defining the cohomology groups of a paracompact space, it is sufficient to consider merely the locally finite coverings
rather than
all open coverings, in the direct limit construction introduced above. For further discussion of the topological properties, see for instance J. L. Kelley, General Topology, (Van Nostrand, 1955).) Theorem 1.
If M is a paracompact Hausdorff space, and if
is an exact sequence of sheaves of abelian groups over M , then there is an enact sequence of cohomology groups of the form
H1(M, .1 )
**
Hl (M
Let ilL = (Ua)
Proof.
the apace M .
0) s H2(M, ) -- ...
be a locally finite open covering of
For each simplex
exact sequence o ---* r(Ia ,
.
a e N(V? )
there is an induced
R. ) -4 r(IaI, 4 ) --*-> r(IaJ, 0 )
;
end since the cochain groups are merely direct sums of the groups r(jaJ, * ) , there follow the exact sequences of cochain groups
, o__> cq(UZ,V
)-> 0 j1> cq(UtDefining
in , 0 ) = *0 (VZ , j ) c Cq(llt
0 ) , these sequences can be
extended to full exact sequences of the form
0--->Cq( homomorphisms
TCq()il,.d) >Cq(711, )--.0. The c and *
clearly commite with the coboundary mappings,
in the sense that qr = 8cp and *8 = Sir ; there results an extensive
commutative diagram of the following sort, in which all the rows are exact sequences of groups.
!
1
o --> Cq-l(?A
1
Cq-1(
Cq-1( 71 , .1 )
sl
51
sj
0 >Cq(11Cq(111,.1) * 0 -- Cq+l( VI ,
Cq
W..) -
Cq+l( ill
I
(IZ0 sj
Sl
51
11Z , ) -- 0
, J.)
.
Cq+l( ?ft, V) -- 0 I
I
Now it follows immediately from an examination of this diagram that for each index
Hq( y1
q
there is an exact sequence of cohomology groups
, g ) -4 Hq( 111 ,1)
Hq(1/i
,)
Hq(M ,
3 ) = Zq( tl
Zq(-n
D) =(f a cq(Vi, 3 )ISf = 0)
,
Hq+l(ZI
S*: Hq(1!1
If
f e Cq( iL, D )
g s Cq(111 ,
j
)
, ) where by definition
3) and .
Mappings
are then constructed in this manner.
is an element for which Sf = 0 , select an element
such that *g = f ; then since $5g = S*g = Sf = 0 ,
by exactness there must exist an element h e Cq+l(71Z, R) such that
[f] a V( U1,
qh = Sg . Define S*[f] _ [h] , where cohomology class of f and
[h] e Hq+l( VI
'k )
)
is the cohomology
class of h .. Of course, it is necessary to observe that and that
[h]
is the 5h = 0 ,
is independent of the choices made in this construction,
namely, the choice of representative
f
in the cohomology class
[f]
and the choice of the element
be left to the reader.
g ; this is straightforward, and will
Finally, another simple diagram chase, which
will also be left to the reader, shows that the resulting cohomology
Hq+1(, )
*
X)
sequence' ... - Hq( Hq+l(
*
Hq(TZ ,
.J) ,,,
Next, consider a refinement
...
*
) ? Hq( 111 , 0 is an exact sequence.
of the covering 2L .
µ: IP --> jjt
There is a similar exact cohomology sequence for the covering it is easy to see that the induced cohomology homomorphism
µ Hq(7l
,
com-
µ
*, **, S* of each cohomologr sequence.
mutes with the homomorphisms (in particular,
; and
Upon passing to the
7 ) C 1(W , 0 ).)
direct limit, there then follows an exact cohomology sequence for the
space M ,
... -- Hq(M,
Hq(M, 0 ) 8 Hq+l(M, ) -
) -2-> Hq(M,
...
Up to this point, the regularity properties of the space M have not been required; but the cohomology groups to be investigated.
Hi(M, 0 )
remain
It will next be demonstrated that, for a para-
compact Hausdorff space M ,
Hq(M, 0 ) = Hq(M, 0
)
, which will
suffice to complete the proof.
It clearly suffices merely to show
the following: given a cochain
f e Cq(21
refinement µf = *g .
µ:
If
-
YL
, 9
)
, there exist a
and a cochain g e Cq( Ir , j ) , such that
Since M is paracompact Hausdorff, and hence normal, there
are open sets Wa such that 9U C U a and the Wa cover M ; and the covering 71t-
can be assumed to be locally finite.
p e M , select an open neighborhood that:
V p
of p
For each point
sufficiently small
(i)
(ii) (iii)
p C Wa for at least one set Wa ;
if P fl Wa # 0 then if
P C Ua ; and
a = (UO,...,Uq) a N(UZ )
necessarily Vp C
and p s Jul
then
,
(so
is the image
pV f(a) p
under * of a section of j over VP
For each set P select a set µ(P) = Up e IQ such that this is always possible by (i), and then exhibits the set as a refinement of
For any q-simplex
Vp C Wp C Up ;
1( _ (P)
,...,V ) e N(7C pq 101 = Vpo n ... n Vpq c W fl ... n w ; since PO pq .
a = (V
po
note that
it follows from (ii) that Vpo C U
Vpo fl W
pi hence that
for each
i
pi
a C Vpo C U
n ... fl U
PO
- Iµal
.
Therefore
pq
Pf(a) = P+alf(PPo....,PV ) = PlaJPV f(Upo,..., Pq) . However it 0 follows from (iii) that the restriction to
of the section
V
f(µe)
0
and this suffices to conclude the
already lies in the image of proof.
Let J be a sheaf of abelien groups over the topological
(d)
space M , and let M .
= (Ua)
be a locally finite open covering of
A partition of unity for the sheaf j subordinate to the covering
1l( is a family of sheaf homomorphisms,
qa: d ---> ,d
.(i)
qa( J p) = 0 for all p e M - U
(ii)
Ea r;a(s) = s for any
Note that since IQ
such that:
;
sej
is locally finite, it follows from (i) that the
sun in (ii) is a finite suet, hence is well defined.
A sheaf j is
called fine if it has a partition of unity subordinate to any locally
finite open covering-of M . F9r example, on any Riemenn surface M , the sheaves are fine sheaves.
for p e M - U. and
ra(p) = 0
and
To see this, recall that for any locally
finite open covering 71L of M there are such that
(
C" functions ra on M Eara(p) = 1 ; for a proof,
see for instance L. Auslander and. R. E. MacKenzie, Introduction to
Differentiable Manifolds, (MacGraw-Hill, 1963).
The operation of
multiplication by the e function ra clearly defines a homomorphism rya
in the sheaf of germs of a or of continuous functions) and
these homomorphis are a partition of unity for the sheaf. Theorem 2.
is a locally finite open covering
If lit = (Ua)
of a topological apace M , and H'(Vt ,.t ) = 0
for all
q > 0 .
a paracompact Hauadorff apace Proof.
Let
(qa)
is a fine sheaf on M , then
Hence, for any fine sheaf A over
M, Hq(M, I ) = 0
for all
q > 0 .
be a partition of unity for the sheaf j
subordinate to the covering Q , and consider an arbitrary cocycle f e Zq(7R , J ) , for
(q-l) simplex
q > 0 .
a = (UO,...,Uq-1) , the induced homomorphiam
sections yields a section since
T6f
For any fixed index a and any
q* (Ua,UO,...,Uq-Z) e r(Ua n IQI, J )
vanishes identically over
can be extended by values
ga(a) = 111
0
IaI
on ;
- Ua n IaI , this section
to determine a section
laf(Ua,UO,...,Uq-Z) e r(IQI,
These sections then define a (q-l) coahain ga a Cq-l( that, for a q-simplex
Tea
v = (UO,...,Uq) ,
"C
Note
q
i
SBa(a) = io(-1) PIQIga(ai) ,
where
ai =
q iE0(-l)ip1alElalr f(Ua,UO,...,Ui-lUi+l,...,Uq
EIa1Sr1f = TIaf(a) , recalling that
and
ga(a)
g = Eager
f
is a cocycle.
This holds for all indices
vanishes identically outside the set U. . is well-defined, since l1L
a ,
The cochain
is locally finite, and
Bg = Ear f = f , in view of the properties of a partition of unity. Thus,aaiy cocycle is cohomologous to zero, and desired.
Hq(111 ,.8) = 0
as
This suffices to prove the theorem.
By applying this theorem, the cohomology groups can be described in the following frequently useful manner.
A fine
resolution of a sheaf J of abelian groups over a topological space M is an exact sequence of sheaves of abelian groups of the form
0 --> -f --> J0
(3)
d0
J1`dl
where the sheaves Ji are all fine sheaves. homomorphisms
J2
d2
...
,
For each of the sheaf
di , there is the induced homomorphism of the groups
of sections over an open subset U C M ,
di: r(U, .1 i) -k r(U, J i+l) ; but the corresponding sequence of these groups and group homomorphisms is not generally exact.
Theorem 3.
If (3) is a fine resolution of the sheaf J over
a paracompact Hausdorff space M , then
Hq(M, J ) _ (kernel dq)/(image dq-1)
Let X i C .11 i
Proof.
for
,
q > 0
be the kernel of the sheaf homo-
morphism 41 ; then the exact sequence (3) can be rewritten as the following collection of short exact sequences: d
d
i>1.
0
A portion of the exact cohomology sequence associated to the first of the above short exact sequences is as follows:
... ---> Hq-1(M, ) Since j 0
)
- Hq-1(M, X 1) --> ON, J) --> ON, J 0) -> ...
is a fine sheaf and
Hq(M, J 0) = 0
.
If
q > 0 , it follows from Theorem 2 that
q = J. , the formula of the present theorem follows
immediately from this exact sequence since Hq-1(M, 1(i ) = HO(M, X 1) = (kernel d1) ; while if
Hq-1(M, j 0) = 0 and
Hq(M, J ) = Hq-1(M, 7'1)
.
q > 2 , then
The exact cohomology
sequence associated to the second short exact sequence above, for the case
i = 1 ,
contains the terms
... -> Hq-2(M, J1) - Hq-2(M, Since
j1
is fine and
'2) -+-. 0-1(m,
q > 1 ,
1) -> Hq-l(M,, J1) ->
Hq-1(M, J 1) = 0 .
If
desired result follows immediately, while if
q > 2 , then
Hq-2X
Hq-1(M,
.1 1) = 0
as well, so
Hq-2(M, K2)
q = 2 , the
Hq(M)
X 1) =
Continuing this process, the desired result follows eventually.
-38-
...
I )
.
In many cases arising in practice, a fine resolution of a
given sheaf appears naturally at hand, and the preceding theorem provides a very useful approach to the problem of calculating the cohomology groups; an illustration of this will be taken up next. But before turning to the example, it may be noted that a fine
resolution can be constructed for any sheaf J , so that the results of the preceding theorem can be applied theoretically to an arbitrary sheaf.
For
any
and any open subset U C M
sheaf
,
let J U be
such that rr o f: U ---> U is
the set of all mappings f: U
the identity on U ; for emphasis, note that the mappings required to be continuous.
f
are not
The collection of these groups J U
together with the natural restriction mappings, form a complete pre-
sheaf over M ; the associated sheaf J
Ka C Ua
so that
UaKC = M ; and define a mapping
maps
is a fine
For given any locally finite open covering Y t= (Ua]
select some subsets
if
will be called the sheaf
It is clear that J
of discontinuous sections of J . sheaf.
*
s E Ka,
Tja(s) = 0
if
qa.
s J Ka .
Ka fl K
J* --> J
for
of M
a
by putting
and nfa(s) = s
It is easy to see that these
qa are sheaf homomorphiams, and they form a partition of unity
for J * .
Flrthermore, there is a natural injection mapping
J --> J *
.
Now, to construct the fine resolution, put ? 0 )*
put
(e)
)l = ( )0/J
; and so on.
For an example which will be of some use later in the present
discussion, consider a connected open subset M of the complex line C .
Introduce on the space
0o M the first-order linear partial
differential operators
Note that the Cauchy-Riemann equations for a complex-valued function f
can be written of/az = 0 ; that is to say, given a function
f E Ca M , then
if and only if of/az = 0 .
f e 61.
The mapping
is a homomorphism from the ring (o M to itself; and
f --> of/az
hence this mapping induces a sheaf homomorphiam
a: 6 °°
T 6'
.
The Cauchy-Riemann c ndition can then be interpreted as the assertion that the kernel of t iia homomorphism is precisely the sheaf 6L of germs of holomorphic functions on M ; thug there is an exact sequence of sheaves
0--> tv --> (p, a> The sheaf
m Co
avw
.
is a fine sheaf over M , as noted earlier; so this
provides part of a fine resolution of the sheaf 61, and raises the question of whether this can be extended further as a fine resolution
of 0 . Let
Lemma 6.
g c e M , and let D be a connected open
subset of the complex line C such that D Then there exists a function whenever
r(z) # 0
f e C M such that
i) C M
af(z)/az = g(z)
z e D .
Proof.
such that
is compact and
Select a
r(z) = 1
for
c
function r
on the complex line
r(z) = 0
z e D ,
only on a compact subset of C .
-40-
for
L'
z e C -M, and
The function
h(z) = r(z)g(z)
for
h(z) = 0
z e M,
for
z e C -M ,
is then a e function on the entire complex line, coincides with the given function
g
on the set D C M , and vanishes outside a
compact subset of C .
Now put
f(z) =
2
.
ffc h(z+5)
d ^d
here the complex differential form notation is used, writing
dt=d6+idn if t_ $+irk, sothat dt^dd= -2idg ^dr), and thus
dt ^ dT
is
times the ordinary plane measure.
(-21)
(r,e) , writing
that in terms of polar coordinates follows that hence
f(z)
,
it
de , and
is clearly a well-defined and a function in the entire Differentiation yields the formula
of
= 23i
21 JJC
=
Now fix the point
.
JJC ah(z+t) dt ^ d
it
=
t = reie
(dt ^ d))/t _ (-2irdr ^
complex line C .
az
Note
271
31 IIC aT (ham)) a5 - dT
Select a disc A centered at the origin
z e C .
and large enough that the function h(z+0 vanishes identically for t e C- A ; and a disc
centered at the origin and of radius
Ae
small enough that A C A .
The boundary of A will be called 7 ,
and the boundary of De will be called
ye ; here,
and
7
of rz
=
lim Ifa
e-j0
a (h(-
ea5
))d
dT
are
ye
Then
circles about the origin, with the positive orientation.
27fi
E
;
and applying Stokes' theorem, and recalling that on the portion
y
of
the boundary of A-6
the integrand vanishes identically, secure
.
that 27ri
Parametrize the circle
of az
yE
27ri
= in f h(z+5) d5 E - o yE Tby writing
t = ceie
,
0 < 0 < 27r , so
2w
of
lira f e-0 h(z+ceie)ide
=
e -> 0
az
27r h(z)ide = 27x1 h(z)
fe=0
Therefore
f(z)
is the desired function, and the lemma is thereby
proved.
As an immediate consequence of this lemma, if
germ of a r function at any point of a
C7 function at the point
p
is the
g
p E C , there exists a germ f such that
of/2z = g
.
Conse-
quently the following is an exact sequence of sheaves of abelian groups:
0 --> 0 --k 4, a
C, --> 0
.
Considering the associated groups of sections, it follows immediately
from Theorem 3 that
r(M,
H1(M,
Hq(M, D )
0
r(M,
az for q > 2
00)
, and
In fact, a slight extension of the above lemma leads to an interesting and useful result.
Theorem 1+.
line, and
g e
M .
Let M be a connected open subset of the complex Then there exists a function f e gyp' M such
that
af(z)/az = g(z) Proof.
for all
z r; M.
Select a sequence of connected open subsets
Dn C M
with the following properties:
Dn
(i)
is compact, and fn C n+l
Un=1 Dn = M ;
(ii)
can be approximated
any function holomorphic in Dn-l
(iii)
uniformly well on Dn-2 C Dn_l by functions holomorphic
on Dn (The last condition is an approximation theorem of the Runge sort; to see that this construction is possible, see for instance E. Hille,
Analytic Function Theory, vol. II, pp. 299 if., (Ginn and Co., 1962).)
Next, by induction on the index n , observe that there is a sequence of functions (iv) (v)
fn
is a
fn with the following properties:
C" function in
afn(z)/az = g
for all
n i
z e Dn
2-11
(vi)
If(z) - fn-1(z)I <
To see this, suppose given the functions
index n > 0 .
hn(z)/z = g(z)
fl,...If n-1 , for some
such that
By Lemma 6, there is a function hn a 0 '0 whenever
In case
n-1 are both C- in Dn-1 , and z e n-1 ; that is,
In case
z e Dn .
is nothing further to show.
for
z s a-2
for all
Ihn(z) - fn-i(z) - h(z) I < 2-n
1 , there
hn and
a(hn(z) - fn-1(z)Xz = g(z) - g(z) = 0
is holomorphic in n-1
holomorphic in n such that for. all
z e n-2 , as a consequence
of the approximation property (iii) above. fn(z) = hl(z) - h(z)
or
n > 2 , the functions
hn(z) - n-1(z)
There exists a function h(z)
n = 0
The function
then satisfies the desired conditions.
Now for any point toy some limiting value
z e M , the sequence
f(z)
(f (z))
n
z c n 9
Indeed, for all points
.
converges
CO
(fl(Z)
E
f(z) = fn+2(Z) +
-
f(z))
m--n+2
Since
If
1(z) - fm(z)l < 2-m for
z s n C Dm-2 ,
m > n+2 , by (vi),
the series is absolutely uniformly convergent in Dn ; and since the
individual terms of the series are holomorphic in n by (v), the sum is also holomorphic.
af(z)Iaz =
Therefore g(z)
f(z)
is e in Dn ; and
in Dn by (v).
This suffices to
conclude the proof.
If M is a connected open subset of the complex
Corollary. line
C , then
H((M, 0)=0 for q> l Proof.
.
This result is an immediate consequence of formula (1+)
and the preceding theorem.
(f)
The cohomology groups of a space with coefficients in a sheaf
have been defined as direct limits of cohomology groups of coverings of that space.
It is natural to ask when the cohomology of the space
can be read directly from the cohomology of some covering; and the answer is provided by the following result.
Let J be a sheaf of abelian groups over a para-
Theorem 5.
compact Hausdorff space M , and
_ (Ua)
be an open covering of M
such that Hl(Ial,
,
)=0
for all a cN(?tj)
and
q>l.
Then
for all q> 0 .
ON, A?) =Hq(111,, .J ) Select a fine resolution
Proof.
d - J0 dc-- Jl of the sheaf ,d
d
r(M, d i)
-
over M r(M,
>J2
Then for the induced homomorphisms
j i+l)
,
it follows from Theorem 3 that
Hg(M, J) _ (kernel dq*)/(image d,* q)
for all
a e N(V?. ) , the cohomology groups
Hq(jal,J )
1
laxly by restricting the resolution (5) to
q > 1 .
Ivl
For any simplex
are determined simi; but since
Hq(lal, £ ) = 0 by hypothesis, it follows that the sequence of sections
0 --4 r(laI, $ ) -a r(IQI, do)
d-°4
...
r(IQI, J1)
corresponding to (5) is actually an exact sequence.
Since further
the cochain groups are merely direct sums of groups of sections over
the various simplicea of N(V. ) , there follows an exact sequence of groups of the form
(6)
o -> cq(
d
cq( Ill ,
0)
d-
cq( ?J2
,
1)
...
The coboundary maps commute with the homomorphisms of the exact sequence (6), so that all of these sequences can be group together in the following commutative diagram:
.
0
(7)
0
0
*
1
r(i, J) - r(M, .d 0) d°-T r(M, d 1)
a
i 0
0
*
1
-- CO(
, d 0)
CO( U?
* C,(Zq
d>
,
*
sl
sl
1) *
d
sl
0 -_> C2 (71i ,
c2 (UZ
) 1
*
, ) 0)
C0( 2
sl
0--> c1(7n,1)->c1(Iq , 10):04cl?n, 11 sl
r(M, J2) -
sj
a
C,(vi, J2 *
Sl
Sl
a c2
1)
d2)
c2 ( 2a ,
j2)
1
All of the rows except for the first are exact, from the exactness of (6); and the measure of inexactness of the first row, in the obvious sense, is the cohomo7.ogy of M .
Since all the sheaves mt i
are fine, all of
the columns except the first are also exact sequences, by Theorem 2; and the measure of inexactness of the first column is the cohomology of the covering 1l(
.
The desired result follows immediately from a
diagram chase through (7); the details will be left to the reader.
As a terminological convenience, a covering
of the space
M which satisfies the conditions of Theorem 5 will be called a Leray
covering of M for the sheaf Corollary 1.
If 24
J
and 2/ are Ll/eray coverings of a para-
compact Hausdorff space M for a sheaf Q , and
g: H -- Ul, is a
refinement, then the induced mapping
µ*: Hq (l
, J) '# Hq (f s J )
is an isomorphism.
Proof. and
The natural homomorphisms u: Hq( ?1L , J) -> Hq(M, H"(M, J )
v: H"('Y'
are isomorphisms, by Theorem 5;
and since v o µ* = u , it follows that necessarily
µ
is an iso-
morphism.
If 7f_ is an arbitrary open covering of a para-
Corollary 2.
compact Hausdorff apace M , the natural mapping
u:
Hl( Vi.
,J
) ---> H1(M, J )
is an injection, (i.e., has kernel 0). Proof.
For an arbitrary open covering
of M there corre-
sponds a commutative diagram (7); the columns are all exact, except
for the first column, but without the hypothesis that VZ is a Leray covering, the rows need not be exact beyond the second place.
How-
ever, a diagram chase shows that it is still possible to conclude that the mapping
Hl(?I?
,
I ) --' Hl(M, J) is an injection; details again
will be left to the reader.
0.
Divisors and line bundles
(a)
One of the main approaches to function theory on Riemann
surfaces involves the study of functions from properties of their zeros and singularities.
The sheaf machinery developed in the pre-
ceding two sections proves quite useful here.
On a fixed Riemann
* surface M , consider the sheaves holomorphic functions and
of germs of nowhere-vanishing
of germs of not identically vanishing
meromorphic functions; in both cases the group structure in the sheaf The quotient sheaf
is multiplicative, and
is called the sheaf of germs of divisors on the Riemann surface. A
section of the sheaf A over a subset U C M will be called a divisor on the subset
U .
Note that a germ of a divisor at a point
p e M , that is, an element of the stalk
is an equivalence
class of meromorphic functions, where two meromorphic functions are considered as equivalent when their quotient is holomorphic and no-
where vanishing; thus an equivalence class consists of all the germs of meromorphic functions having the sane order (the same zero or
pole) at the point p .
In this sense, divisors merely furnish a
description of the zeros and singularities of meromorphic functions. In the case of a single complex variable, the sheaf an alternative and much simpler description; and this simplicity is
one of the distinctive differences between the function theory of
one and of several complex variables. For any germ f e equivalence class of vp(f)
of the function
mp*
, the
in Sp is described uniquely by the order
f f
at the point p ; the stalk ZIP = 7, / (/p
-J.8-
is therefore naturally isomorphic to the additive group of the integers.
(Recall that p(f-g) = p(f) + Vp(g) , so that the multipli-
cative structure in m p corresponds to the additive structure of
the orders V(f) e Z .) To describe the topology of , = 7t/ O*
,
recall that such a quotient sheaf is always topologized by defining the images of sections of
'YY(*
as a basis for the open sets of
over a basis of the open sets of M ,
'
.
Now for any open subset U and
any meromorphic function f e T(U, flt*) , the image set in J is the divisor of the function
f ; and the important thing to note is that
the order of a meromorphic function f set of points in U .
is non-zero only at a discrete
Thus an open set in .4 will consist of an
integer associated to the points of an open subset U C M , in such a manner that non-zero integers appear only for a discrete set of
points in U . as follows.
It is thus clear that the sheaf J can be described
To any open subset Ua C M associate the additive group
61a of all mappings
V: Ua -- Z
such that
V(p)
0
only on a
discrete subset of Ua ; the group structure is of course the pointwise addition of the functions.
If Up C Ua , the natural restric-
tion of such functions from Ua to
pPa:
UA
is a group homomorphism
This defines a complete presheaf over M , and
the associated sheaf is just the sheaf A of germs of divisors. This latter description will generally be used henceforth. that, from this description, it is obvious that jS
Note
is a fine sheaf
over M ; the details of the verification will be left to the reader. As for notation, divisors will generally be denoted by German
script d , namely, " . To describe a divisor
e r(u, 4) ,
it is of course sufficient to give the orders (the integers) at only
those points of the discrete subset of U where the order is nonzero; thus divisors will be written
vi.pi,
where
Vi a Z,
pi a U,
M*)
For a meromorphic function f e r(u,
and
Ui pi C U is discrete.
, the divisor of f will
be denoted by 9'(f) ; thus
(f) A p
e U Vp(f)'p
where the sum can be restricted to the discrete subset of U consisting of points at which vp(f) # 0 . - (fg) _ $ (f) + , (g) ; and that ,5 (f)
function f
0 .
Note that is not defined for the
The divisors over U can be given a partial order-
ing by defining
j
= £i Viepi
> 0 provided Vi > 0 .
Note then that holomorphic functions
f
over U are characterized
by the condition that 9 (f) > 0 ; and more genernnal]y, A (f) > ,9(g) if and only if
f/g
is holomorphic.
Divisors N such that n4 > 0
will be called positive divisors.
The mapping which associates to a meromorpluc function f
its divisor ,t9 (f) is just the natural homomorphism from the sheaf 7M *
,,9
:
711* -> ,(9
to its quotient sheaf; this can be described by
writing the exact sequence of sheaves
(1)
where
i
is the natural inclusion mapping.
(The notation
0 will
always be used for the trivial sheaf, whether the group structure of
the stalk is considered additive or multiplicative.)
Corresponding
to this sheaf sequence over M is the familiar exact eohomology sequence, in which appears the homomorphism
,g *: r(M, ?'i.*) -+ r(M, AQ ) . An element ,9 a r(M, A5.)
is a
divisor defined over the entire Riemann surface M ; while an element in the image of
is the divisor of a meromorphic function
,a
defined over all of M .
That there exist non-trivial divisors
defined over all of M , or equivalently that
r(M, L
) # 0 , is
completely obvious; but that there exist non-trivial meromorphic
functions defined over all of M , or equivalently that
r(M, M) / ©,
is far from trivial, is indeed one of the basic existence theorems of the subject.
Thus the question of whether or not the mapping
is onto is one of some interest.
In a special case, the answer is
immediate.
Theorem 6.
(Weierstrass' theorem)
If M is any connected
open subset of the complex line 0 , the following is an exact sequence of groups:
o-->r(M, Proof.
r(M,fl#)-r(M,
The exact cohomology sequence corresponding to the
exact sheaf sequence (1) begins as follows:
o - r(M, Q *) ---> r(M, %t*) - r(M,
1
H1(M, cD *) -
therefore to prove the theorem, it suffices to show that II'(M, t ) = 0 .
Recalling the exact sheaf sequence
0 -- Z -4 (t -
-- 0,
when
a (f) = exp. 21r i f
there is an associated cohomology sequence, which includes the segment
...
?
I
HI(M, B ) -- H1(M, Nov by the collorary to Theorem 4,
that 11(M, B *) _ H2(M,Z) . dimensional manifold, as desired.
--+ 112(M. B) .
H'(M, )
= 11(M, 9) = 0 ; so
But since M is a non-compact two-
H2(M,Z) = 0 , and therefore
H1(M, & *) - 0 ,
(Seethe topological appendix for a discussion of the
assertion that H2(M,Z) - 0 .) Remarks.
The corresponding theorem holds for an arbitrary
non-compact Riemaam surface M ; the only result needed is that R1(M, Q ) = R2(M, ®-) - 0 .
(For the proof) see for instance
R. C. (kenning and H. Rossi, Analytic Functions of Several Complex
Variables, p. 270, (Prentice-Hall, 1965).)
The theorem implies that
an arbitrary divisor on M is the divisor of a global meromorphic
function on M .
The Weierstrass factor-theorem gives an explicit
representation for a function with the prescribed divisor; (see for instance L. Ahlfors, Complex Analysis, p. 1,57, (McGraw-Hill, 1953) ) For compact Riemann surfaces the preceding theorem does not
hold at all; we shall see eventually that
(M, A) # 0
.
An
investigation of the precise extent to which the theorem fails will
be one of the main topics of consideration. A few trivial observations and further definitions are in place here.
Recall that the
sheaf & of germs of divisors on a Riemann surface was noticed above to be a fine sheaf; hence by Theorem 2,
H'(M, 0-) = 0 .
Therefore the exact cohomology sequence associated to the exact
sheaf sequence (1) has the form
(2)
-> r(M, ,) -H1(M, £*) '*> H'(M,7 *)--0.
o -4 r(M, 4 #) - r(M, 7 #)
The quotient group
;(M)
r(M, A )/ J*r(M,
is really the measure of the extent to which Theorem 6 fails to hold;
and in terms of that group the exact sequence (2) can be written
0 -- A(M) -> Rl(M., 4 *) -a Rl(M, %t*) -a 0
(3)
As a matter of terminology, the group
group of divisors on M .
r(M, At) will be called the
Two divisors
called linearly equivalent, written
A91,
J2 a r(M, ,(Q)
will be
2 , if their difference
n¢1
is the divisor of a meromorphic function on M , that is, if
JZ -
n¢2 = AQ (f)
for some f e r(M, WL)
This is an equivalence
relation, is indeed the equivalence relation corresponding to the homomorphism
,.
in (2); in particular, the image of
group of divisors linearly equivalent to zero.
*
is the
The group A(M)
is
called the divisor class group on M , and is the group of linear equivalence classes of divisors on M .
The exact sequence (3) will
later permit a rather complete description of the group A(M) , and thus settle the question of the extent to which the Weierstrass theorem holds on compact Riemann surfaces.
(b)
In the further discussion of these questions, one is led in
a very natural manner to investigate a special class of sheaves. introduce these sheaves, consider the group appeared notably in the above discussion.
h1(M,
)
, which
This group will be called
the -grroup of complex line bundles over M ; and a cohomology class
9 E H (M, 0 *)
To
will be called a complex line bundle over M .
(The terminology arises from an interesting geometric interpretation
which can be given to the elements
g c H1(M, Q*) ; this geometric
interpretation is totally irrelevant to the purposes at hand, although it is not uncommonly injected into the discussion of these topics, and will be ignored here.
The interested reader is referred to
F. Hirzebruch, Neue Topologische Methoden in der Algebraischen Geo-
metrie, (Springer, 1956).) For any complex line bundle
9 e H1(M, QL*) , select a basis
ll
for the open sets of M , and a cocycle
{Ua)
(L ) e Z1( U(,
)
representing that cohomology class; since bases are cofinal in the
open coverings of M , there always exists such a representation. elements
Ea¢
The
are holomorphic, nowhere-vanishing functions defined
in the open sets Ua fl u, , and the cocycle condition asserts that CO
(P)
. gP7(P) _ gy(p) whenever p e Ua n UP f1
set Ua a
associate the group J a = r(Ua, s-)
functions in Ua .
of holomorphic
To each inclusion relation UP C U a associate
the group homomorphism pPa:
function
To each open
7 .
a -> Jp , which associates to a
f e Ja = r(Ua,.) the function pra(f) a ,1 P
= r(UP, LT )
defined by
(PO4) (P) = gPa(P) f(P) for p e UP C Ua . Note that whenever Uy C UP C Ua and f e,da , then f(P) _ g,a(P) f(P) = (p7
= Ey0(P) that is,
p7Ppsa = pya .
Therefore
) (P)
(pyP(pO
))(p) _
for all p e Uv ;
{U1 , J ,pap)
is a presheaf
over M , which is readily seen to be a complete presheaf; the associated sheaf is called the sheaf of germs of holomorphic cross-sections of the line bundle
g
, and will be denoted by
9-Q)
.
It is a
straightforward exercise, which will be left to the reader, to show
that the sheaf d-(9)
is defined independently of the choice of co-
cycle representing the cohomology class
g
, that is, that the sheaves
constructed in terms of two cocycles representing the same cohomology class are isomorphic sheaves.
Since the above presheaf is complete, there is a natural
identification r(Ua, c (g)) a Ja = r(Ua, 9-) . that an element
f e r(M, to (g))
fa e r(Ua, (Q
where
)
It is then clear
corresponds to a collection
{fa} ,
and
fa(p) _ o(p) d' (p) whenever p e Ua fl u, ;
(4)
these sections of
will also be called holomorphic cross-sections
of the line bundle
Note that the set of all such sections has
the structure of a complex vector space, as well as just that of an
abelian group; and that (9 = ( (1) , where 1 E H'(M, a *) is the trivial line bundle.
The construction just described could have been carried through just as well for the groups J a = r(Ua, 711) ; the homomorphisms {
p.a are well defined, as above, and the collection
J, po}
is again a complete presheaf.
The associated sheaf
will be called the sheaf of germs of meromorphic cross-sections of the line bundle
g , and will be denoted by 9i (9)
f e r(M,1, (a))
correspond to collections
.
The elements
{fa} , when the functions
fa are now meromorphic functions satisfying the relations (4); such sections will also be called meromorphic cross-sections of the line bundle
9
.
,d a = r(Ua,
Or, in the same manner, using the groups
there arises a sheaf G (t;) which will be called
the sheaf of germs of The sheaf 6 °'(9)
cross-sections of the line bundle
CO*
g
.
is of course always a fine sheaf, an observation
which will be of use later.
the order of f at a
For a cross-section f 6
point p
is a well-defined integer P(f) ; for defining
vp(f) = vp(fa) functions
(fa3
when p e U« , and recalling that the meromorphic satisfy equations (4) where
are holomorphic
1a3
P(fd) when-
nowhere-vanishing functions, it follows that P(fa)
ever p e Ua n u, .
Note that for any section f which is not
identically zero, the order is non-zero only on a discrete set of
points; hence to the section f there is associated a well-defined divisor
(f)
F EMVv(f).r Then
called the divisor of the cross-section f e r(M,
r(M, ® ( )) C r(M, ?j W) appears merely as the subgroup of meromorphic cross-sections of the line bundle
g
having positive divisor,
that is, r(M, al (0) _ (f a r(M, 'l (01 A (f) > 0) . One further general remark of importance is that, for any line bundle
g e H1(M, m *)
,
g = 1
(the trival line bundle) if and
only if there exists a cross-section f e r(M, B (1)) (f) = 0 .
For 4 (f) = 0 means that the functions
such that (fa)
are
holomorphic and nowhere-vanishing in
U(I , and from equation (4) they
form a zero-cochain in
having
so that
Co(U(
,
g = 1 ; convdrsely, if
a *)
9 = 1 , then
I
as its eoboundary, r(M, ®(g)) ffi r(M, ® )
and this contains the non-zero constant functions.
Of course, in a
1
parallel manner, for a cohomology class
g e Hl(M, 7Y1*) ,
if
9 = 1
and only if there is a cross-section f E r(M, %(g)) which is not identically zero.
Recalling the exact sequence (3), every element (y.*)
g e H1(M, 'VA*)
can be represented by an element of
111(m,
l1(M, 9t*) = 0
therefore one can assert that on any Riemann surface, if and only if, for every line bundle
g e H1(M, 6 *)
The vanishing of the cohomology group
H1(M, 7)(*)
;
,
r(M,ryn(g)), 0.
is therefore
equivalent to the fundamental existence theorem for Riemann surfaces, namely, the theorem that every line bundle has a non-trivial (not identically vanishing) meromorphic cross-section; and this is also of course equivalent to the assertion that every line bundle is the line bundle of a divisor on the surface. To any divisor ,R a r(M, 41)
exact sequence (2) a line bundle
there is associated by the
8*JT a
R'(M., (9*) , and hence also
the sheaf 6(8*A ) of germs of holomorph c cross-sections of that
line bundle; to simplify the notation, set ®. (J ) = Ql(8*4- ) . has another interpretation of interest as
Nov the sheaf S (f& ) well.
To the divisor A associate a subsheaf
fined as follows.
For any point p e M let
(( g )p = (f e'j{1pleither f = 0 or and put OM (A ) = p Ok
(9m ( ) C 9 de-
EM
(lm
j (f) > J near p
)p . It is clear that each
(,9 )p C rM p is a subgroup, and that £M (J ) C 1 is an
open subsets hence dM lemma 7 The sheaves isomorphic.
is a well-defined dubsbeef of rill and
(TM (,9 )
are canonically
It is necessary to examine the homomorphism
Proof.
the exact sequence (2) a bit more closely. A a r(M, IQ.)
in
For the given divisor
it follows from the exactness of the sheaf sequence (1)
,
that there are open sets
(U.)
of M , and
forming a covering
da defined in the various sets Ua , such that
meromorphic functions J (da) _ J IUa
8 '
Then in each intersection Ua n UP the function
.
gCO = d/da is holomorphic and nowhere-vanishing; and the collection of all such functions define the line bundle
(g00 )
g = 6#,9 a H1(M, 0 *)
Co (V1
.
(The functions
(da)
, 1M *) which maps onto the zero-cocycle
and the functions
of the cochain
(gap)
form a zero-cochain in
j a Z°( 11l ,
£!) ;
form the one-cocycle which is the coboundary
(da) ; recall the proof of Theorem 1.)
To each germ
f e OM( nQ ) p C N and to each open set Ua containing p
asso-
ciate the germ fa = f/da e 'fr p . Since n9 (fa) = A (f) - j (da) _ J_(f)
- A > 0
near
p , the germ fa will necessarily be holomorphic
at p ; and if p e Ua
11 Up , then fa - f/da = f- F,,a p/dp = gcp ° f,. There-
fore the functions (fa) define the germ of an element in 0 (g )p
)p . This defines a mapping from (( W ) to if
(,' ) ,
which is readily seen to be an isomorphism, and thus completes the proof.
Since the sheaf 9
is defined in terms of holomorphic
functions, it is the easier to handle analytically and will play the greater role in the present discussions.
However, in view of the
isomorphism V (,D) '= aM(A ) , any results about the sheaf carry over to results about the sheaf Q7,
(J )
and this re-inter-
pretation frequently leads to interesting statements.
For exanple,
r(M, &( Q_ )) functions
f
j (f) > ,,¢ a
is the vector space consisting of those meromorphic
defined on the entire Riemarni surface M such that ; and the dimension of this vector space is a number of
some interest.
For a compact Riemann surface M the spaces
(c)
11(M, Q-(g))
are finite-dimensional complex vector spaces for all dimensions and any line bundle
g e H1(M, (*) ; in fact
q > 0
Hq(M, S (g)) = 0
for
q > 2 , as we shall see in the following section, so it suffices to prove the finite -dimensionality only for
q = 0
and
1 .
In demon-
strating this, it is convenient to topologize the cochain and cohourology groups and apply a few simple results of the theory of topological vector spaces.
At this stage there is a choice to make,
since one can either topologize the full spaces of cochains (as Frechet spaces), or pass to certain subspaces of cochains which admit simpler structures as topological vector spaces (namely, as Hilbert spaces).
The latter approach has been selected here, to minimize
prerequisites; but the arguments are basically the same in either approach, and in higher dimensions as well, following H. Cartan and J. P. Serre, (C. R. Acad. Sci. Paris) 237(1953), 128-130)-
First, let U C C be a connected open subset of the complex line, with
that
z = X+ iy
dx ,. dy
as the complex coordinate function on U , so
is the standard Euclidean plane measure in U.
.r0(U, C- ) = {f a r(U, I)IffUlf(z)12dx
thus
ro(U, a) C r(u, B')
dy < .o)
Define
;
is a vector subspace, which will be called
the space of square-integrable analytic functions in
U.
For any two
functions f,g a r0(U, 6L) , it follows innnediately from the Cauchy-
Schwarz inequality that (5)
(f,g).U = ffUf(z)g(z)dx . dY
is a well-defined positive definite Hermitian inner product on the space
r0(U, 6 )
TO(U, (1) , in terms of which
space, (that is,
PO(U, 9 )
is a Hilbert space in all but complete-
ness); the norm in this space is given by f e PO(U, 61), at
z0
z0 E U , and A(z0,r)
and such that
IIfIIU
= (f,f) U .
is a disc of radius
Now if r
centered
A(z0,r) C U , note that
r
If(zo) I <
(6)
is a pre-Hilbert
IIfIIU
Yir
z = z0 +
(For writing
peie
a A(z0,r),
0 < p < r , the Cauchy integral
formula reads
hence 2
f(z0) = jr
Jp=O 1e=0 f(z0+pe3A)pdpde
therefore
If(z0) I <
ffa(zO,r) If(z) IdX
-r2
<
s as desired.)
2
r
dy
(ffA(z0,r) 1 dX .. dY)1(ffA(zO,r)If(z)I2dx
dY)i
IIfIIU
Now it follows readily from this that
complete in the above norm, hence that
r0(U, Q )
r0(U, I )
is
is a Hilbert space.
To see this, consider any Cauchy sequence of elements
For any compact subset K C U , select a number
0(z,r) C U for any point
fm(z) - fn(z) I < the sequence
1
r7r
r
(n (z))
convergent on K .
11f
m
e PO(U1
f
r > 0
-)
so small that
z e K ; then by (6),
- fn""U
for all points z e K , so that
is uniformly Cauchy on K , hence is uniformly
The limit function
f(z) = limn fn(z)
holomorphic in U , and is the limit of the sequence
is then in norm as
fn
well. If U C V C C are two connected open subsets of the complex line, the restriction mapping
pU: T0(V,
.) --> r0(U, 9- )
is a bounded linear operator between these Hilbert spaces.
Further,
if U is compact and U C V , this restriction mapping is even a compact operator, (that is, takes any bounded subset of to a subset of
T0(U, (9
)
having compact closure).
merely a restatement of Vitali's theorem; for if sequence in (fn(z)I
r0(V, OL )
r0(V, 61)
This is actually
(fn)
is any bounded
, it follows from (6) that the functions
are uniformly bounded on the compact subset U C V , and
hence by Vitali's theorem there will be a subsequence which is uni-
formly convergent on U . To apply these considerations to the problem at hand, select
g e H1(M, 4*)
a line bundle
over the Riemann surface
M.
Let
(f _ (U«) be a coordinate covering of M in terms of which the line bundle
g
is representable; that is, let Pt be an open covering
of M such that:
.
(i)
1
each set Ua is contained within a single coordinate neighborhood;
(7) (ii)
has a representation cocycle
the line bundle
(tCO) e ZI(V\ , V#) For any simplex and
IaJ
a e N(V(. ) , then,
r(ja
,
a(g)) = r(1vI, 61 )
;
is contained within a single coordinate neighborhood.
Selecting any fixed coordinate mapping for the set
, one can
r0(jaj, &(E)) C r(JaI, 61(g))
then introduce the subspace integrable sections of
IaI
CL (9)
over
I aI
spaces, over all q-dimensional simplices
of square-
; and the direct sum of these a e N( V( ) , is a subgroup
C Cq(Vl , Q (g)) which will be called the group of square-integrable cochains of
llt with coefficients in
Co(g)
Since the coboundary operator involves only restrictions and Finite
summations, it is clear that
B: C"( ?A , 111(8)) --> CO+l(
introducing the kernel Z0( Ul , QL (g)) C Cg( Vt, (Y (g)) of the coboundary mapping, the corresponding square -integrable cohomology
groups are defined by
H (?!1 ,
(g)) = zo(vl , m(g))I s Co-l(u,
Notice that the inclusion
1: Cg( UI ,
(Q (g)) - Cq( M , a (g) )
induces a homamorphism
ice: Ho( U(,
(g))
Hq( 7JZ, (D (g))
Lemma 8.
Let Vt = (Ua)
and
= (Va)
be open coverings
of M , where 7A is a refinement of W such that Ua C Va ; and suppose that both coverings satisfy condition (7). dimensions
q = 0
and
q = 1 ,
Hq(MO(a))
HH(Yt, (p (a)) = Hq( ?1Z, (D (a)) Proof.
Then, for
First, observe that the homomorphism
i*: HH()Jl , ()1(g)) -a Hq( VI , (P (a)) is an injection (isomorphism into) for dimensions
q = 0, 1 .
This is entirely obvious for q= 0.
For the case
q = 1 , select a cocycle
suppose that
(fa4)
that is, that
(fCO) a Z1(U1
,
, and
(P (a))
is cohomologous to zero in H1(Ut , (Q (a))
(fa) a Cl('Ui , (Q (a)) ; it suffices
fa - f13 for
f ao =
to show that each fa
,
is square-integrable over Ua .
For any point
p e IUa , select an open set UP such that p e UO ; thus in Ua n u, ,
fa = fad + fo .
However, the functions
and
fP
are
fad
both square-integrable in the intersection of Ua with some open
neighborhood of p well.
in M ; hence
fa
is square-integrable there as
Ua is compact, finitely many of these neighborhoods
Since
cover aUa , and so clearly fa
is square-integrable in U. , as
desired.
Now, since each simplex
a
of either covering 2( or V lies
within a single coordinate neighborhood, and on
lal
there is the
isomorphism (a) = ( P, it follows from the corollary to Theorem 4 that Hq(a,,
(a)) = 0
for all dimensions
q > 1 ; hence, by
Theorem 5,
Hq(1 , (Q (g)) = Hq(lf , (Q (a)) = Hq(M, (V (a))
dimensions
q > 0
, for all
Indeed, as in Corollary 1 of Theorem 5, the
homomorphism refinement simplex
of
µ*: Hq(y , (9 (1)) -- Hq( ,
µ:
l1Z ---> It
Q! (g))
is an isomorphism.
a e N(Jjl ) , its support
I a I
induced by the
Note that for any
lies within a compact subset
'µ(a)' ; so clearly the mapping µ* can be factored as follows:
*
q(1f Since
°> Ho(1n is an isomorphism, and
µ* = i*µO
dimensions
Hq(vt ,
q = 0, 1 , it follows that
is an injection in
i
* p*
d
is then an isomorphism,
and that concludes the proof. Theorem 7. e
If M is a compact Riemann surface and
H1(M, G *) , then the cohomology groups
Hq(M, LV (g))
dimensional complex vector spaces for dimensions
Proof. Let UJ = {TIa), `W _ (V(,}, open coverings of M , where Ua C Vu,
11t
q = 0
q = l .
is a refinement of I' such that
and 1f' is a refinement of )ZO
a
and
and )i/ = {W«} , be finite
such that V. C Wa ; and
suppose that all three coverings satisfy condition (7).
simplex
are finite-
For any
in the nerve of any of these coverings, the space
r0(Ia1, (9(k))
has a Hilbert space structure; thus the square-
integrable cochain groups are Hilbert spaces, since they are finite sums of Hilbert spaces.
The coboundary map being continuous, the
square-integrable cocycles are also Hilbert spaces; but one cannot immediately say the same thing for the cohomology groups, since the space
500-1(VZ
,
(T (a))
of coboundaries is not yet known to be a
closed subspace.
First consider the case of dimension HO(
q = 0 .
Since
ZO( lf(, (Q (g)) , this cohomology group is a Hilbert
The refinement mapping
space.
operator µ*: Hg ()r from Lemma, 8 that
p
s
p: 7 ---> 10 induces a bounded linear
It follows directly
m (9)) ---> H6 (VI
is an isomorphism of complex vector spaces,
hence by the open mapping theorem is an isomorphism of Hilbert spaces. On the other hand, as a consequence of Vitali's theorem as noted earlier, the mapping
is a compact
t*: r(Va, G (g)) ---> r(Ua, CL
operator; and since the cochain groups are finite sums of these spaces,
These two
the cohomology homomorphism is also a compact operator. observations taken together show that
Hg( Z(, 3 (a))
is a locally
compact Hilbert space, hence is necessarily finite-dimensional. Next, for dimension
CO( 1k
,
q = 1 , consider the Hilbert space
(g)) + Z10 ( ( , (Q (a)) .
Applying the coboundary mapping
to the first factor and the refinement mapping
µ
to the second
factor yields a homomorphism
(s,µ): C0(V7 , c W) + Zl('W , 6 W) ---> I_Ua
(8)
,
(Q W)
It follows from lemma 8 that this mapping is suriective; for since i
:
H'-('e ,
0 (c)) ---> '_( 1 ,
(g)) is an isomorphism, any cocycle
of the covering VI must be cohomologous to a cocycle of the covering If
.
On the other hand, as noted above, the refinement mapping
µ
alone is a compact opertor; so applying the zero mapping to the first factor and
(9)
µ
to the second factor yields a compact operator
(o,4: co(1R , C' (g)) + Zo(W , y (f))
The difference
(8,µ) - (0,µ)
l_(
of the two mappings in (8) and (9) is
Just the coboundary map 8: CO( jf(, QI (a)) - z o-( uj
the
, a (g))/S(o(
desired result, that the quotient ZZ (Vz
is
LO (g))
a finite dimensional vector space, is then an immediate consequence of the following lemma.
Let X and Y be Hilbert spaces, and p: X --> Y
Lemma 9.
*: X - Y be bounded linear operators such that
and
jective and
yr
is sur-
qi
is compact. Then (ep_i): X T Y has closed range
of finite codimension in Y . Proof.
(For some of the general properties of linear
operators required here, see N. Dunford and J. T. Schwartz, operators, vol. I, (Interscience, 1958).)
Y --> X be the adjoint mappings to then
Let
p and
Y -- X , respectively;
is a one-one mapping with closed range, and
cp
compact mapping.
For suppose that
any bounded sequence of elements of K , so that yr#
is a
yr
Note first that the kernel K of the map
.is a finite-dimensional subspace of Y .
Since
p*:
Linear
-
q (yn}
is
(p* -yr*)(yn) = 0
is cong,act, then after passing to a subsequence if neces-
sary, the sequence #(yn) = yr4(yn)
rp
will converge; and therefore
yr (yn)
converges.
Since
cp*
is one-one and has closed
range, it is a homeomorphism between Y and its range, hence converges.
(y
n
}
This shows that K is locally compact, hence finite-
dimensional.
Now, factoring out by K , we may assume that one-one mapping; and we then show that (yn)
Consider a sequence ((P* -
**) (yn) --a x
.
If
cps -yr#
-yr
is a
has closed range.
of elements of Y such that (yn)
has a bounded subsequence, then as
above we may suppose that
**(yn)
CP(Yn) =
(Yn)
converges; but then
converges, so again yn converges
to an element y , and (p _**)(y) = x YnIIYnII-1
the elements
yn =
consider
IIynII
0 . Again we may suppose that
converge, so that
an element y' .
If
; these elements have norm 1, and
(cp(Yn) = **(yn)
.
Now
11Y' II
p*(y'n)
and hence
= 1 , and
this contradicts the assumption that
((P* -
p -$
yn
converge, say to
4r*) (y') = 0 = ((P* - yr*) (0) is one-one, and this
second case cannot occur.
Tp conclude the proof of the lemma, since closed range, the same is true of Y' = Y/(cp- p) (X)
p': X -3 Y'
X - Y' space
Y'
the proof.
p -fir
is a Hilbert space.
T - $
has
; so the quotient space
Now p induces a mapping
which is surjective, and i
induces a mapping
which is compact; and since clearly p' = *' , the
is locally compact, so finite dimensional, which concludes
§5.
Differential forms and Serre duality.
(a)
Some familiarity with differential forms, at least the defi-
nitions and most elementary properties, will be assumed here.
(As
references for this material, among other books are: L. Auslander and R. E. MacKenzie, Introduction to Differentiable Manifolds,
(McGraw-Hill, 1963); and S. Helgason, Differential Geometry and We shall only
Symmetric Spaces, Chapter I, (Academic Press, 1962).)
consider the case of two-dimensional manifolds M , the following notation.
and shall adopt
The sheaf of germs of complex-valued
r will be denoted by S
differential forms of degree
r ;
C`*
thus the
vector space of such differential forms over an open subset U C M is just the space
r(U, (r) .
erential form of degree 0
,
note that
r = 0 ,
since a diff-
f 0 = (o .
is a function,
0
coordinate neighborhood U
c2 _
For the case
tol 9
O + DO
Within any ,
and
(considered as sheaves of abelian groups); for if
,
are local coordinates in the neighborhood U ,
point of U ,
then any element
rp e
p
x,y
and p e U is any
can be written uniquely in
the form q, = f dx +g dy for sane germs f, g e £ P -C B , and any element r a G_ P can be written uniquely in the form r = h dx for some germ h e 10
F .
dy
Of course, e r = 0 whenever r > 2
These are clearly all fine sheaves over M The operation of exterior differentiation is a sheaf homomor-
phism
d;
, r
r+l
;
and there arises the exact sequence of
sheaves (the de Rham sequence)
-68-
0-bc--> f0 a_> gl a>e2-->0,
(1)
recalling again that M is
2-dimensional.
Since the sheaves
F r
are all fine, it follows immediately from Theorem 3 that
Hq(M,C) 9 ker d*/im d# q q-1
where
dQ:
r(M, fc) -> r(M, C q+l)
for all
q > 0 ,
is the homomorphism of sections
induced by exterior differentiation; this is of course de Rlzam's Theorem.
For the particular case
q = 2 ,
it takes the form
2
g2(M,C)
(2)
l)
ar(M, P
If M is a compact H2(M,c) S C ;
2-dimensional manifold, then as is known,
(see the topological appendix for further remarks).
Using (2), this isomorphism can be described explicitly as follows.
For any differential form c e r(M, E2) , well-defined complex number; the mapping
the integral cp ---> f1Mcp
ffMcp
is a
is clearly
rKC,2)
a linear mapping from
onto the complex numbers.
-> C ,
cp = d*
so that
it follows from Stokes I theorem that j fMcp = j jMd.* - 0 , this mapping yields a well-defined linear homomorphism
If
r(M, t 2)/dP(M, P
which must indeed be an isomorphism since both sides are
one-dimensional complex vector spaces.
Now suppose that M has a complex structure; and select some coordinate mapping z« = p e M .
Writing
xx' + iya
dza = dd a + idya
in an open neighborhood of a point and dza = dd x - idya
furnishes
a new basis for the e P-module
e 1 I
so that
epa 1Pdza+epdza.
(3 )
If z = x, + iyo
is another complex analytic coordinate mapping
defined in anopen neighborhood of p ,
za = f(z)
it follows from the Cauchy-Riemann
zP
is a holomorphic function of
then since
equations that
dza = fap
dz,
`tea
J.
a3
and so, p
ar
GpdZa=Gp
pdzPCpdzP
It follows that the splitting (3) is intrinsically defined, that is, is independent of the choice of local coordinate mapping.
110 p
0
p
dz
a
e0 dz the splitting (3) becomes p a ,
and to" p
tp'0 + Fp'l
e1a Since
5 P
F p
a ,
of uniformity write to r,s
Thus
F
a
d y
s
Fp
2 =
11
p
p
,
dza % dza ,
J.
we shall for the sake
and similarly F
0 =
p
0,0 p
denotes the sheaf of germs of complex.valued differen-
tial forms of total degree and
Writing
basis elements
r + s
dEa ;
involving
of course here
r
dza
basis elements r = 0
or
1 ,
but
the situation in the case of several complex variables is quite parallel.
Ma terms of the splitting (31), the exterior derivative d:
F0
> F1
d = a + a
= F 110 + e0,1 can be split as a direct sum
where
a:
F 1,0
610,0
and
za a + iya ,
For a coordinate mapping f(z)
,
F 010
and a function
> to0,1 . f(x,y)
secure
,
of dz + () dz of xa a+C`a°Fa a
of
a
where
a/aza
and
are the linear differential operators intro-
O aza
duced in §3(e), (page 40); this is a straight forward verification which will be left to the reader.
of =
Tz-
dza
,
a
It thus follows that
of = a aza
dza
In a parallel manner, for a differential form degree
1
w = fadza +gadza
it follows that
g
dza Adza + - dza
dw
A dza = 6(f
a)
+ ()(gadza)
aza
a
The de Rham sequence (1) then splits as follows:
It is interesting to look at the separate pieces of this
.71-
of
splitting of the de Rham sequence more closely.
First of course
there is an exact sheaf sequence (the Dolbeault sequence)
0 --> d---- £ 0,0 a 60,1 > 0 .
(4)
The exactnees follows immediately from Lemma 6, and (4+) is really merely an invariant form of the exact sequence of page 4+2.
Since
all the sheaves F r, are fine, it again follows from Theorem 3
that Hl(M, e) = r(M, FO'1)/ar(M, FO'0) and that for
q > 2 ;
Hq(M, 19-) = 0
this is Dolbeault's Theorem, and is an invariant re-
statement of equation (4) of page 4+2.
Next, there is an exact se-
quence of sheaves of the form
0>
(5)
where a
1,0 6L
in (5).
C 9
d1'° ---> S1'0
1,0
> ei'l > 0 a
is defined as the kernel of the hamomorphism
Introducing a local coordinate mapping
neighborhood of appoint p g M
za
in the
a germ of a differential form
,
fa(za)dza a `-p10 belongs to the subsheaf
Q 1'0
if and only
if
0 ° aq) °
'
)dF ^ dza
that is, if and only if the function tion near
p .
The sheaf d 1,0
fa(ze)
is a holomorphic`func-
is therefore called the sheaf of
germs of holomorphic differential forms of type
(1,0)
,
or also
the sheaf of germs of Abelian differentials; a section of this sheaf
is a holomorphic differential form or an abelian differential.
Note that for an abelian differential p ,
dp _ Zkp +
p = 0 ;
thus every abelian differential is a closed differential form. These forms can be introduced in several complex variables as well, but in the higher-dimensional cases they are not automatically closed forms, which makes for further complications.
Now select a complex line bundle
(b)
cross-sections of the complex line bundle
the parallel construction yields sheaves
and recall
400(g) of germs of
the construction given in §l+(b) for the sheaf COD
e) ;
g e H1(M,
g
' It is clear that
.
t r's(g)
of germs of
differential forms which are cross-sections of the line bundle
Cm g
.
be a basis for the
(To carry out the construction, let
?ll. - (Ua)
open sets of M such that each Ua
is a coordinate neighborhood and
that the line bundle
t
can be represented by a one-cocycle
To each open set
(E _) a
group J a = r( a,Fr's) ;
and to each inclusion relation a C U
associate the group homomorphism PaP: differential form
pap 'pp
Ua associate the additive
pp a r(UU,
a r,s)
'Ot
P ->
,j
a which takes a
to the differential form
a r(%, a e r's) defined by (Paa pO)(P) = gas(p) * cp,(P)
for
p e Ua C UO .
This defines a complete presheaf, whose associated sheaf is the sheaf
E
r,s(g)
.)
These are clearly fine sheaves.
-73-
The ordinary exterior derivative cannot be applied to these sheaves to obtain an analogue of the de Rham sequence, since exter-
ior differentiation does not commute with multiplication by
lap .
However there does arise an analogue of the Dolbeault sequence.
if
cp e e
r, s (g )p at a point p e M , then
cp
is represented in
each coordinate neighborhood Ua containing p by a germ a differential form of type
then a
pp
(r,s)
at
p ;
Now since the functions
For
of
cpa
and if p e U n U
a
are holomorphic,
6P that is satisfy aF = 0 , it follows that acpa = FP acpP i fore (acpa) - acp is a well-defined element of C r, s+l(g )p .
thereThis
leads to a sequence of sheaves (the Dolbeault-Serre sequence)
(6)
o --
e°'°w a> P011(t) ---- 0
In a single coordinate neighborhood this sequence reduces to the Dolbeault sequence, hence (6) is an exact sheaf sequence.
The
following generalization of formula (4) of §3(e) then follows trivially.
Theorem 8.
If M is any Riemann surface and
is a line bundle over M ,
t H H1(M,
*)
then
H1(M,e(t)) a r(M,E0 1(t))/ a rX e°'°W) Hq(M O(t)) = 0
Proof.
for q> 2
Since the sheaves £
r,s(t)
are fine, the Dolbeault-
Serre sequence (6) is a fine resolution of the sheaf
& (g)
desired result then follows immediately from Theorem 3.
1
the
There is one result of prime importance, which is the basis of the further study of compact Riemann surfaces; indeed, the core of the analytic side of the theory consists of this result (and Theorem 4 or its analogues).
Theorem 9.
(serve's Duality Theorem).
Riemann surface, and M .
be any complex line bundle over
t e H1(M, B *)
Then the vector spaces
Let M be a compact
and H0(M,
H1(M, 6 (g))
m1,0(t-1))
are
canonically dual to one another, hence have the same dimension.
The proof of this theorem will be given in the next section, following Serre (Un Theoreme de DualitLe, Comm. Math. Helv. 29 (1955), 9-26).
In fact, the techniques in the proof of the theorem are of
quite a different sort than the applications, and the reader who is willing to take this theorem on faith can omit the proof entirely and pass on to the applications.
It is perhaps of interest to indi-
cate briefly here just what the duality actually is, though. q) e r(M, F 0'l(t))
and y e r(M, e" O(t-1))
if
C" cross-
are any two
sections of their respective line bundles, then note that their exterior product
q) ^ y e r(M, F1'1)
.
For in any coordinate neighbor-
hood Ua these sections are represented by a differential form of type
(0,1)
at a point
and a differential form
p e Ua n U
60(p) pi(p)
and
yi(p)
for
a(P)
yP(P)
p e Ua n U0 ;
(1,1)
The products ,
and
Ta(p)
therefore the products
define a global differential form of type M .
(1,0) ;
and
these differential forms satisfy Ta(p) _
are then differential forms of type Ti(p)
of type
ya
cpa
Since M is compact, the integral
(1,1)
p(,
^
" ya
*a(p)
Ta - *a
on the manifold
((P,*) - ffM cP
(7)
-*
is a well-defined complex number; then (7) defines a bilinear mapping
r(M, F°'1(g)) x r(M, F1'°(g_1)) --> c . Now if
q) e a r(M, £ 010(8)) C r(M, e0
1(0) ,
so that
p = of where
f e r(M,F°'°(g)) , and if * e r(M,1'0(g_1))C r(M, 91'0(8.1))
,
so that a* = 0 , then (T,*) = j"M of
* - ffm a(f*) = ffM d(fi) = 0 3
for since M is compact, and since fir a r(M, a l,0) C r(M, Fl) it follows from Stokes' theorem that
ffM d(f*) _= 0
.
Therefore the
pairing (7) leads to a pairing
r(M, (0'l(g))l a r(M, a°'°(g)) x By Theorem 8, as always,
r(I`'I, m1'0(g-1))
->
H1(M, ®(g)) 2 r(M, £0 l(g))/ a r(M, F°'°(g))
H0(M, &10((1)) = r(M, O 1'0(g-l))
.
g.
s
and
This therefore
describes a bilinear pairing
H1(M, & (g)) X
H0(M, dl'O(g_l)) --> C .
The assertion of Serre's theorem is that this is a dual or nonsingular pairing, hence that the two spaces are dual vector spaces.
The
spaces are then isomorphic as complex vector spaces; but whereas the
above duality is canonical, the isomorphism is not.
For most of
the applications, what is required is merely that the two vector spaces have the same dimension.
It should be remarked in passing that the Serre duality theorem holds for higher-dimensional manifolds as well, in the sense that Hq(M, B(g))
and Hn-q(M,
& n,O(g-l))
are dual on any
compact complex manifold M for any integer
2n-dimensional
0 < q < n .
The greater
strength of the theorem in the case of Riemann surfaces lies in the fact that all questions can be expressed in terms of the zero-dimensional cohomology groups in that case; in higher dimensions, one is faced with the problems of handling the cohomology groups Hq(M,0 (t))
(c)
for
q - 1,2,...,[7]
as well.
The Serre duality theorem can be expressed without explicit
mention of differential forms, by observing that the differential forms involved can be considered as cross-sections of line bundles themselves.
This introduces a particularly useful line bundle,
defined intrinsically on any Riemann surface as follows. 7f(.= (Ua)
Let
be a complex analytic coordinate covering of the Riemann
surface, with coordinate mappings
za:
Ua > C .
The coordinate
are complex analytic local hameomorphisms
transition functions fap
between open subsets of C ,
all p e Ua
n
U0
the intersections
such that
za(p) = fap(zP(p))
for
Now introduce the functions K, defined in
Ua M UP by
-77-
Kc(P) = (f
since the functions
(ZP(P))l'1
;
are holomorphic and nowhere-vanishing, faP
the same is true of the functions then
p e Ua A UP n U
KaP .
Furthermore,
if
ay(zy(P))
za(P) =
so
,
that by the chain rule
[fay(zy(p)!l-1
= (f' (fP7(zy(p)))
Kay(P) =
K P(P)
thus (KaP) c Z1(ZJt
KPy(p)
*)
1
defined by
The element K E H1(M, CO)
this cocyele is called the canonical line bundle on the surface. Note that this bundle is independent of the choice of covering, since
it can be constructed for a maximal covering of the surface M .
Now consider the sheaf 4 tials on M .
1,0
of germs of Abelian differen-
coordinate mappings
za:
Ua -> c ,
an element
611,0
a
dza = fP
so that
dz, ,
fa = dzWdza
if
dza
a p c Ua
fP = KUP
the coefficients (K)
.
fa
in each then
n UP , fP ;
thus
can be considered as elements of the sheaf
61'0 ° 6 (K)
This then establishes an isomorphism
a completely parallel manner there is an isomorphism for any line bundle
is
q) e
represented by a germ of a differential form Ta
coordinate neighborhood Ua containing p
with
= (Ua) ,
In terms of a coordinate covering
g
,
where the product
Kg
.
In
(91'0(8) _ ® (Kg)
is taken in the
(9*)
group
(M,
.
Considering
C" rather than holomorphic sheaves,
-78-
there is the isomorphism
sheaf $ 60°(Kg)
0,1(g)
j,
(1,O (g)
9 p 0,0(Kg) = t "(Kg)
.
(The
can be considered as isomorphic to the sheaf
when K denotes the complex conjugate of the canonical
bundle, in the obvious sense.)
In these terms, the Serre duality theorem can be restated as follows.
Theorem 91. e
H7(M,(*)
tor spaces
Let M be a compact Riemann surface, and
be any complex line bundle over M .
H1(M, 0(g))
to one another.
and H0(M, & (Kg-1))
Then the vec-
are canonically dual
Proof of Sarre's duality theorem.*
§6.
The proof will require some rather simple results about distri-
(a)
butions; for the benefit of those not too familiar with distributions,
we begin with a brief but self-contained review of those results which will be needed.
U of the complex line C , with the coordinates
to subdomains z = x + iy .
The first part of this discussion will be restricted
ort of f is the
For a function f e C U , the s
point set closure in U" of the set
(z C Ulf(z) # 0) ; the support
is thus a relatively closed subset of U , which will be denoted supp f .
The subset of
0 U consisting of those functions having
compact support will be denoted by derivatives of functions
o rp U
.
To simplify notation,
U will be denoted by
f e
2f
Vf (V11v2) (f) _ = D D
6V1+V
,
where
V
(V1,v2)
3xV1ayV2
Definition. T:
o
A distribution in U
is a linear mapping
U --> C , such that for every compact subset K C U there
are constants M and n with the property that
(1)
If the integer
IT(f) I < M
E
sup IDVf(z)1
when
supp(f) C K
Vl+V2 < n z e K n
can be chosen independently of K , the least
possible value is called the order of the distribution.
all distributions in U
The set of
is a linear space which will be denoted by
kU
This section can be omitted on first reading, or omitted altogether by readers willing to take the Serre duality theorem on faith. '
As an example, suppose that
g
is a Lebesgue measurable
function in U which is integrable over any compact subset of U ; there is an associated distribution T9
defined by
for f e
Tg(f) = f f(z)g(z)dx .. dy U
a o
U
It is clear that this is actually a distribution, indeed, a distri-
In particular, the space a U of infinitely
bution of order zero.
differentiable functions is thus naturally imbedded as a subspace
C U C n
U.
As another example, to any point
associated a distribution
a e U there is
8a , the Dirac distribution centered at
a , defined by
8(f) = f(a)
for
C
f e
00
o
U This is also clearly a distribution of order zero; and one sees thus that the space of distributions is properly larger than the space of locally integrable functions.
If T e , U and gee U , the product
is the
gT e
distribution defined by (gT)(f) = T(fg)
for
f e
o
10 r°U
It is obvious that this is a distribution, and that it has order at
most that of T that whenever
U X xU --
T
if
is a distribution of finite order.
g,h e 6 U , then Tgh = '`U
Note also
thus the product
is compatible With the ordinary product of func-
tions, on the subset
U C X
U
.
However, this cannot be extended
to an associative product on the full space '1 U
-81-
of distributions.
If T e ) U , the derivatives of that distribution are defined by
for feo9U It is also obvious that these derivatives are distributions, and that
aT/ax
and aT/ay are distributions of order at most n+l if
T
is
a distribution of order n ; higher derivatives are defined inductively. Also, the linear partial differential operators
a/az
and
a/az
introduced in §3(e) can be applied to distributions as well as to functions.
It should be observed that this definition is compatible
with the usual notion of differentiation on the subspace
For if f e o 0 v
and
g e I U, then
a(f)=-Tg _ - f
Since
supp(gf)
theorem that
0D C 2 U
=-Ug(z)afz)dxAdy -E(g(z)f(z))dx .. dy + I f(z) U
a
z)
dx A dy
is a compact subset of U , it follows from Stokes'
fU a(gf)/ax dx ,. dy - 0 ; and therefore
6T
a ('Rg(f) _ f f(z)
(z)
dx
dy = Tag/6x(f)
U as desired.
The same result holds also for
derivatives as well.
a/ay and for all higher
This observation can be used to give a meaning
to derivatives of arbitrary locally integrable functions, considered as distributions; and in fact, all distributions arise in this manner.
Note that Leibniz' rule holds for differentiation of the product of a
e function and a distribution.
If V C U are two aubdomain$ of the complex lines then clearly o
extended to a function U -V .
f e o e U by setting it identically zero in
Then any linear functional
linear functional pWT on
if T e x U
defined on
' o
V by restriction.
o
(Q°
U
defines a
In particular,
The restriction mapping is thus a homomorphism
---A X V ;
W C V C U .
T
it is obvious that the restriction pWT is a distri-
bution in V .
pW:
f e 0 4 v can be
G Y C o U; for every function
and it is clear that ppW = pW wherever
Thus if / = (U)
is a basis for the open sets in the
topology of C , then the set (vj , 2U,pw) defines a presheaf over C ; the associated sheaf will be denoted by N , and will be called the sheaf of germs of distributions over © . The presheaf of distributions is a complete
Lenmta 10.
presheaf; hence there is a natural identification for any open subset
U.
Let U be a fixed open subset of 0 , and let
Proof. (Ua)
F(U, X) _ )(U
be an open covering of U .
Recalling the definition of a
complete presheaf (page 19), there are two assertions to be proved. First, suppose that
U
pU US - pU UT
S,T a )(U
are distributions such that
for all Ua ; then it must be shown that
By
S = T .
a
passing to a refinement of the covering if necessary, there is no loss of generality in assuming that
(Ua)
is locally finite.
(ra)
be a, C" partition of unity subordinate to the covering
with
supp ra
compact for each a .
f = £ of ; since
supp f
For any
f e o
Let (Ua)
U , write
is compact, only finitely many terms of
this series do not vanish identically. and
supp(raf) C Ua
it follows that
,
Thus since the stmt is finite,
S(f) - S(£j(xf) - a,S(raf)
= aT(raf) = T(Earaf) - T(f) , which suffices for the desired result.
Second, suppose that Tae NU a
for all Ua n u,
PUa fl UP,UPT
PUa n
are distributions such that
;then it must be
shown that there is a distribution T e x U such that
pU UT = Ta 0:
for each a .
Again assume that
(Ua)
is a locally finite covering,
and select a subordinate partition of unity
(r(.)
f e o C U can be written
supports; so that any
the sum is a finite sum.
with compact
f = Faf^, where
Define a linear functional
T:
o coo --'> C
by setting T(f) = Zcja(raf)
On the one hand, note that that
f e o e U.
pU UT = To ; for if
supp f C U0 , then since
T(f) = EaTa(raf) = Ea(PUa
for
n
f e o
supp(raf) C ua n u,
is such
it follows that
UPIUaTa)(raf) = a(PUa n UP,UPTP)(raf)
'On the other hand, observe that
aT13 (raf) = TP(f) .
U
T
is actually
a distribution.
For given a compact set K C U and a function
f e o 6 v with
supp f C K , since the sets
are compact and the
Ka = K fl supp ra C ua
Ta are distributions, it follows that
IT(f)l < Z ITo;(raf)l Ea%
E
Vl+V2 < na the set of indices
a
sup IDV(rof)(z)I ; 2 E Ka
in the above summation is actually finite, and
depends only on the set K , so the above inequality clearly reduces
to an inequality of the form (1), when n = max na and M is suitably chosen.
Then
T
is a distribution, and the proof is concluded.
The support of a distribution T e K
is defined to be the
U
set of points in U which have no open neighborhood to which the
restriction of T
is the zero distribution; the support will be de-
noted by supp T , and is clearly a relatively closed subset of U Note that for a function
g e
further that w h e n T e x U
supp g = supp Tg ; and note
U ,
and
g e
s u p p (gT) C supp gflsupp T
U
Then multiplication of distributions by a C" partition of unity defines a partition of unity in the sheaf
that is, the sheaf
of germs of distributions is a fine sheaf.
(b)
In a sense, the Cauchy-Riemann conditions hold for distriBefore turning to
butions as well as for differentiable functions.
the proof of this assertion, a few further simple properties of distributions are required. Lemma 11.
g(z,t)
Suppose that
C X 8 , and that for any number support of
g(z,t)
fixed compact set
t
as a function of K .
Then if
T
is a
in an open interval z
Proof.
is a distribution in an open
of
For any point
t e I
is a
C"
and any h # 0 , note that FE(z,t+h)- g(z,t
L As
t
I .
h[Tg(z,t+h) - Tg(z,t)] = T
(2)
i C R the
alone is contained in a
neighborhood U of K , the function Tg(z,t) function in the interval
function in
Coo
h -. 0 , for a fixed value
t , the function
J' [g(z,t+h)- g(z,t)]/]
as well as its partial derivatives of any order with respect to x
and y , converge uniformly on K ; and their supports are always contained in K .
It then follows immediately from the definition of a
distribution that the expression (2) approaches Therefore
Tg(z,t)
T[ag(z,t)/at]
is a differentiable function of
.
t , and Its
derivative is
H
Tg(z,t) = TIag- (aE 1
Repeating the argument, the function function of
t
in the interval
Lemma 12.
I .
G(z,9)
is a
C"
function in C X C , Then if
is a distribution in an open neighborhood U of K ,
T If G(z, )ds ,. dt = f f TG(z, t)dt .. d
(3)
C
C Proof. z
is thus e as a
Tg(z,t)
supp G C K X L where K,L C C are compact sets.
and that
T
Suppose that
.
Note that
Ife
d1
with support contained in K , and that
tion of
t
is a r function of
TG(z,t)
is a r func-
by Lemma 11, and has its support within L ; therefore
both sides in (3) are well defined. ffe G(z,t)ds
dd are all
COO
The Riemann sums for the integral
functions of
z
with support con-
tained in K ; and these sums, as well as their partial derivatives,
of any order with respect to x and y , converge uniformly on K Then (3) follows again immediately from the definition of a distribut ion.
With these properties out of the way, the Caucby-Riemann conditions for a distribution read as follows; recall that the holo-
morphic functions can be considered as imbedded in the space of
distributions, by associating to a holomorphic function h the distribution
Th
Theorem 10.
such that
T
If
is a distribution in a subset U C C is a holomorphic function in U
aT/az = 0 , then T For any constant
Proof.
let Ue C U be the subset
e > 0
is at least
of U consisting of points whose distance from C - U e ; and select a
r(z) = 0
Iz(< e/2,
for
r(z) = 1
in C such that
function r = re
C**
f c o G U with
Then to any function
for
supp f C Ue
(z() E . associate the
function h(z) = 2
this function is clearly
tai
ah az
=
For a fixed point at
C
bb
Cr everywhere,
supp h C U ,
and
I (t) dt . dd= ff of z+) I (t) d5 .. dT .
ff of z+ C
dt .. d
ff
a
az
z e C let
L.
be a disc of radius
5
z ; then 22ti
2h
az
of z+)
lim
If
= lim
ff
=
5 -40 C-
dt ,. dT
a (f(z+S) r Q
dt
dT)
5 -4 0 C- AB lim f f
5->0 C-ii5
f(z+t)
= 2ai f(z) - f f f(z+S)
as in the proof of Lemma 6.
-lJ dt A dT `` fr-
1
dt .. dT
Note that the function
centered
2[i
a
r
I
0 is actually
everywhere, since
C`0
r(S)/S
for
#0 ,
for
=0,
is holomorphic for
Then write
0 < Ifl < e/2 .
f(z)
ah z)
+ ff
f(zts)ge(S) d ^ dS
C
2Z
izz)
+ If g6(5-z)f(S) dS .. dd C
Since
aT/az = 0
,
it follows that
(h) = 0 ; and so,
T(?h/az)
BE
applying Lemma 12, if - ff Tg6(5-z)f(5)d5 .. dd
C That is to say, the restriction of T which is a
aT/az = 0 and
T
is a
C00
function of
to the set t
Ue .
is the function
by Lemma U. Since
C' function in UE , it follows from the
ordinary Cauchy-Riemann conditions that T in
UE
This holds for any value
holomorphic throughout U
is a holomorphic function
s > 0 , hence
is indeed
T
thus concluding the proof.
It is possible to continue in this vein, securing a fine
resolution of the sheaf
61- by sheaves of germs of distributions,
paralleling the discussion of §3(e); in particular, for any subset
MCC,
a
xl(M, 0) = r(M, X )
Ai r(M, x)
.
We shall not need this, so pursue the matter no further here.
In order to extend the discussion of distributions to Riemann
(c)
surfaces, it is first necessary to discuss the transformation pro-
Suppose that U,V are subdomains of the
perties of distributions.
h: U - V
complex line C , and that
is a
C" homeomorphism.
*
The mapping h
induces a linear mapping h
°e V ---> e U , defined
by h (g) = geh , the composition of the two functions It is clearly of interest to extend the mapping
g
h
and
h* to a linear
h*:
mapping
e U C XU
T9 E XU .
h*:
XV ---> XU , recalling again the natural imbedding which associates to a function
For this purpose, define a linear mapping
xV - xU
by (h T)(f) = T[(f°h-l)Jhl]
(4)
where
g e 6 U the distribution
T e xV,
the mapping h .
f e P U , and Write
Jh
z = x+ iy
is the Jacobian determinant of
for a point in U and
for a point in V , so that the mapping is of the form
Then for any functions g e
V and f e
o
_
+ irk
t = h(z)
(oD U note that
(h Tg)(f) = T9[(f°h-1)Jhl] =
=
f f g(S)f(h l(5)) a(xon) dt A dr) C V
f f g(h(z))f(z) dx z E U
dy
Th*(g)(f) .
Therefore the mapping
h*
on X V , when restricted to the subspace
V C Y, V , coincides with the earlier definition of subspace.
and
h*
It is a straightforward verification that when
T e N V then
on that g E 6ji V
h*(9T) = h*(g)-h*(T)
(5)
and that when
k: V
-> W is another
;
C" homeomorphism and
T e 9(W then (koh) T - h*(k T)
(6)
The details will be left to the reader.
Now let M be a Riemann surface, with a complex analytic coordinate covering za = fa13 (zP)
(Ua,za}
.
(UJ,za}
and coordinate transition functions
A distribution T
on the coordinate covering
is defined to be a collection
za(Ua) = Va C (
the various subsets
(Ta)
of distributions on
such that for each non-
empty intersection Ua n u, C M , (7)
fa43(pza(Ua fl u ),V.a) = 'z,(Ua n U,),VPTP
Two distributions (UJ,za)
T
and
T'
on coordinate coverings
and
(Ua,za)
are called equivalent if they define a distribution on the
union of those coordinate coverings; that this is an equivalence relation in the proper sense is a consequence of (6).
An equivalence
class of distributions on coordinate coverings of M is defined to be a distribution on the Riemann surface.
The sheaf 9{ of germs
of distributions is then a well-defined sheaf of abelian groups on
M , and by lemma 10 the global sections of the sheaf X are precisely the distributions on the Riemann surface.
imbedding of the
The natural
C" functions in the distributions exhibits
goo C /< as a subsheaf, in view of the remarks above.
for any line bundle
I e Iio(M,
Furthermore,
the corresponding sheaf ?(
of germs of distribution cross-sections of the line bundle
I
can
be constructed, paralleling the discussion of §4(b); for that construction merely requires that the multiplication of local sections
of x by e functions be well defined, and so using (5) there are no difficulties.
Adopting the
Details will be left to the reader.
notation of §5(c), we shall write X"°() = X (Kg), x(KK9) , where
and
K E H1(M,
is the canonical
(g
bundle on the surface.
The description of the sheaf ^ can be put in a somewhat more invariant form as follows.
covering of M , with Va = za(Ua) functions
mapping
za =
fad
043
q) E r(M,
and with coordinate transition
is
x
K
be a coordinate
(zo) ; note that the Jacobian determinant of the
a(xa,ya)
where
(Uaza)
Again let
dza
2
dzO
= IKpa12 ,
is the canonical bundle of M .
Suppose that
tl'l) = r(M,
e°°(IK12) , and that T e r(M, 9(
corresponds to a family of functions
f*a(cpp) = I Koal2cpa
cpa E r(va, a W)
q)
such that
in each intersection Ua fl u, # 0 ; and
corresponds to a family of distributions
f'(Ta) = T,
Thus
Ta a P(Vc,, )( )
T
such that
(The precise
in each intersection Ua n u, # 0 .
meaning of the phrase "in each intersection Ua n U. " is of course given by equation (7); the additional notation is cumbersome and of no great necessity, so will be dropped.)
if
supp-tcp c Ua n u, , then
(f'Ta)(Q = Ta['Poef0a)Jfl
=T 1
a
a(cP0)Jf (P
so that
T((p)
is well defined.
More generally, let
be a C"
(ra)
partition of unity on the manifold M , and suppose that
supp tp
is
a compact subset of M ;then define
(8)
T(W) = Ea Ta(ra(p)
There are at most finitely many values of a for which Ta(ra(pa) 10 so that only a finite scan is involved; and as above, the sun is readily
seen to be independent of the choice of partition of rUty. this, suppose that
(sa)
(To see
is another such partition of unity.
Za Ta(ratpa) _ 7-
Zap
C43
supp(rasocpa) C Ua fl u, .)
Then
Tp(rspcpp) = T, TO(socpp) , since P
Therefore the elements of
r(M, x) can
be envisaged as linear ftmctionals on the subspace o (M, x`1'1) r(M,
01'1)
consisting of differential forms of compact support.
In particular, if
bution, then
c
g e r(M,
B00)
Tg((p) = ffM gtp .
and
Tg
is the associated distri-
It is thus clear why the distributions
must be considered as linear functionals on the space of differential forms of degree 2 rather than on the space of functions; for this
integral has an invariant meaning only inthis case.
In a precisely
parallel manner, for any line bundle
the space
r(M x (g-1)) or(M,
I e H1(M,
(k*)
can be viewed as a space of linear ftmctionals on
J'111(f)) ; the details will be left to the reader.
The latter
assertion can of course be rewritten, so that the space r(M,
or(M,
can be viewed as a space of linear functionals on or yet again, the space
I(M, x l,0(9-l))
viewed as a space of linear functionals on o (M,
r0,1(0)
can be if
the distributions rather than the functions are required to have compact support, the corresponding assertions clearly hold; so that
0r(M,
(g-1))
can be viewed as a space of linear functions on
"'(I)) , and so on.
r(M,
If the Riemann surface M
then o (M, V *(j)) = r(M, C O*(j))
and
is compact,
or(M, x (j)) - r(M, '((g)).
One can, in fact, say considerably more.
To simplify this
discussion, suppose that M is a compact Riemann surface, and that (Ua,za)
is a finite coordinate covering of M with the property
that the mapping
za can be extended to be a
C`0
homeomorphism of
an open neighborhood of Ua into C ; and put Va = za(Ua) C C as For any integer n > 0 and any element
before.
f = (fa) a r(M,
put
2 °°(g))
pn(f) - Z
(9)
sup
E
a Vl+v2 < n za a Va
with the notation as in §6(a).
so defined on
r(M, Q"(g))
IDVfa(za) I ,
It is clear that the functions
are norms, in the sense that:
W pn(f) = 0 , with equality holding only when (ii)
pn(f+g) = Pn(f) + Pn(g) ; and
for any constant
c e C .
pn
defined by this family of norms; so
pn(cf) =
(iii)
Introduce on
r(M,
r(M,
(p°
f - 0 ;
Q`D(g))
0(9))
the topology
is a topological
vector space, and a basis for the open neighborhoods of the zero element consists of sets of the form for some
n )
.
(This makes
(f a r(M,
r(M, 0 "(g))
C°°(WIIPn(f)I < E
into a Frechet space;
cf. C. Goffman and G. Pedrick, First Course in Functional Analysis,
Prentice-Hall, Englewood Cliffs, N.J., 1965.) A linear functional T: r(M,
C is continuous in this topology if and only if
there are an integer for all,
n and a constant
f e r(M, c "W) .
c
such that
IT(f)1 < cpn(f)
On a compact Riemann surface M , the space
lermna 13.
r(M, 21l()) is the space of continuous linear flanctionals on the topological vector space If
Proof.
(Ta) a r(M, X 1'l(I)) , then T determines
T =
r(M, C°(g)) by (8), where
a linear functional on
suitable partition of unity.
tinuity of T
r(M, 000(g))
is any
Cr.)
Ta are distributions, the con-
Since
follows immediately from a comparison of the definition
of a distribution, (equation (1)), and the definition of the norms pn ,
(equation (9)).
linear functional on
Conversely, suppose that
T
000(e))
be extended to a section f e r(M,
by putting
zP a zP(Ua fl U0) C VP , extended to a function in
Ta: o
.
for
g-'(zoffoo(zo))
fa(za) = f(za) ; and putting
linear functional
can
Any function f e oa v a
r(M,
it zero on the rest of V
is a continuous
VP by setting
Setting Ta(f) = T(f)
defines a
C ; and it again follows readily
V a
from a comparison of (1) and (9) that Ta is a distribution in Va . f e o r V
Note that if (f
supp f c za(ua n u) , then
has
where
(f) = T(fefCO)a) Koa12) _ IK
f(f,(z,)) .
is defined by 9-1a(za)f(za)
9-(zafa(za) = J
therefore
,
Since
ga(zes) a
it follows that
1$aj2Ta , so that
T(g)
[Ta) a r(M,
g e
V
Pa(za)g(fpa(za))-
1T(f) ;
.(i1))
foes]
Now for any f e r(M, follows that identifies
C°°(g))
and any partition of unity
(ra)
it
T(f) = T(F raf) = Z P(raf) = Va(raf) ; this therefore
T with the section
to conclude the proof.
(Ta) a r(M, a(
and serves
With these preparations out of the way, let us now turn to
(d)
the proof of the Serre duality theorem.
Recall that the statement
of the theorem is as follows. Theorem 9.
(M, 0*)
e
spaces
Let M be a compact Riemann surface, and
be a complex line bundle over M .
H1(M, V(g))
and H°(M, 6110(g-1))
Then the vector
are canonically dual
to one another. Proof.
Introduce the complex vector spaces
A = r(M, 80,O(g)) ,
A T B , as in §5(b).
and the homomorphism Theorem 8 that
B - r(M, a °'1(g)) ,
It follows from
B/aA , and from Theorem 7 that this
H'1(M, (4(g))
quotient space is a finite-dimensional complex vector space.
Further introduce the norms pn given by (9), in both of the spaces
A and B ; this makes these spaces into Frechet spaces, and it is obvious that the homomorphism in terms of this topology.
a closed subspace of B . of the mapping
A/K
a: A -? B
is a continuous mapping
Note further that the image
aA C B
is
(To see this, let K C A be the kernel
a , so that K
is a closed subspace of A ; thus
is also a Frechet space, and the mapping
continuous linear mapping with trivial kernel.
a: A/K Let
B
is a
L C B be a
finite-dimensional subspace of B which is complementary to S(A) = a(A/1{) .
The Product
(A/K) X L
and the mapping a + is (A/K) X L -4- B
is a continuous mapping.
its image is all of
is then a Frechet space,
given by
This mapping also has trivial kernel, and
B ; so by the open mapping theorem, it is an
isomorphism of Preehet spaces.
Since
(A/K) X 0 C (A/K) X L
is a
closed subspace, it then follows that ( + i) (A/K x 0) = aA is a
closed subspace of B , as desired.) means that
inherits from B
B/
The fact that a is closed a topology as a Prechet space,
which must of course coincide with the usual topology of a finite dimensional vector space.
The conclusion of interest here is then
that any linear functional on the quotient space necessarily continuous since
B/dA , which is
is finite-dimensional, leads to
B/aaA
a continuous linear functional on B which vanishes on the subspace
Now, purely formally, let
A*, B* be the dual spaces to
the topological vector spaces A and be the dual homomorphism to kernel
a .
B ; and let a : B* -> A*
It then follows readily that the
K* C B of the homomorphism
is the dual space to
B/dA .
For on the one hand, any linear functional T on B/dA determines a continuous linear functional T on as noted above; and thus
T e K* then T
B which vanishes on dA ,
T e B* , indeed T e K .
Conversely if
defines a continuous linear functional on B
which vanishes on
aA , hence a linear functional on
B/BA
.
Then, to conclude the proof, we need merely appropriately
identify the spaces AB*, and the homomorphism
It follows
immediately from lemma 13 that A* = r(M, x 1'1(g-1))
s
B = r(M, u 1,0(t-1))
;
and recalling the definition of the derivatives of a distribution,
it is clear that V _ -a , where applied to distributions.
a
is the familiar operator as
Now B/aA = 111(M, 0 (f))
is therefore
dual to the kernel of the mapping -a: r(M, x 1'0(1-1))
r(M,
but by Theorem 10 this kernel is precisely 6]l'0(9-1))
H0(M,
r(M, (9
1,0(1-1)) _
, and thus the proof is concluded.
An examination of the proof indicates that the duality is indeed that described in §5(b); details will be left to the reader.
Riemann-Roch theorem.
§7.
(a)
Before turning to the Riemann-Roch theorem itself, it is
necessary to introduce a fundamental invariant associated to W complex line bundle, its Chern class or characteristic class.
This
is actually the first step in the classification of complex line bundles, in the sense of providing a detailed description of the group
xl(M, 4*)
over a compact Riemann surface M .
Recall the
exact sequence of sheaves (cf. §2(d))
(9 e
0 --> z where the homomorphism
0
was defined by e(f) = exp 2itif .
e
The
associated exact cohomology sequence includes the segment
H1(M,z) Since
(1)
- > H1(M, B-) -. H-(M, A*) __> H2(M,Z)
--->
H2(M, )
.
-) = 0 by Theorem 8, this sequence can be rewritten
H2(M,
o --> H1(M, iff )/H1(M,Z) T H1(M,
*) --k H2(M,Z)
o.
The coboundary homomorphism in (1) will be called the characteristic
c: H1(M,
homomo1rphism g e ii(M)
(4*)
.
9*)
H2 (M, Z)
, the image
c(g)
; for a line bundle
will be called the characteristic
class or Chern class of the line bundle
g
.
The sequence (1) goes
a good deal of the way towards describing the group of line bundles in more detail; there remains the problem of investigating the group H1(M, (7 )/H1'(M,2Z) , and this will be tackled in a later section to
complete this point. In a sense, the Chern class measures the topological proper-
ties of the line bundle
g
sheaf <°"*
C
of germs of
.
To make this precise, consider the
complex-valued, nowhere-vanishing
functions on M , the group structure being multiplicative. sheaf
19*
is a subsheaf of
a sheaf homomorphism
ro
The
, the natural inclusion defining
is e T 10"* ; and also there is an exact
sheaf sequence of the form 0 --> Z - Q" -s> 4:"* -> 0 , paralleling the above sheaf sequence in the analytic case.
Alto-
gether, these two exact sequences can be written as parts of the commutative diagram of sheaves and sheaf homomorphisms, as follows.
0->Z -> Q e> 0*> 0 0 --> Z ->
(°"
e-> r*
>0
The cohomology sequences can then be written together as a commutative diagram o. groups and homomorphisms of the form
H-(M, m)->H'1(M, &*)e>
i*j
") T Hl'(M, 4"*) -T H2(M,Z) - H2(M, 6 ")
Hl(M, The homomorphism
°1
i*1
c
in the second line is the parallel to the
characteristic homomorphism in the first line; but in the second line c
is an isomorphism, since
H1(M, d-) = H2(M, 0-) = 0
because
*)
these sheaves are fine.
such that
Now if
c (g) = 0 , the image
g e H1(M, 0
is a line bundle
i *(9) e H l(M,
/°
ci*(9) = c(g) = 0 by commutativity; but since at that level,
obvious, so that
i*(9) = 0
c(g) = 0
.
c
*)
will satisfy
is an isomorphism
The converse holds as well, as is if and only if
is the topological form of the line bundle
i*(9) = 0 g
,
.
Now
1*(g)
so that we may say
that the line bundle is topologically trivial precisely when it has
zero Chern class. (IOP) a Zl( i*(9) = 0
,
Selecting a representative cocycle
9 *)
for the line bundle
is Just that there exist nowhere-vanishing C"
fa defined in the various sets 9CO(p)
, the condition that
g
functions
Ua and such that f(p)/fa(p) =
for p c Ua fl u, , and the condition that
g = 0
there exist holomorphic nowhere-vanishing such functions
is that fa ; this
observation may help to clarify the above discussion.
Since we shall henceforth assume that M is a compact twodimensional manifold, it is known that H2(M,Z) = Z ; the Chern class
c(g) a H2(M,Z)
of a line bundle
g
can thus be considered
This identification
as an integer, under the above identification.
of the Chern class as an integer can be made more explicit as follows. The class
c(g)
can be considered as an element of the group
H2(M,C);
for either apply the cohomologyr homomorphism 1?(M, Z) ---> H2 (M, C) derived from the inclusion mapping X C C of sheaves, or recall from the universal coefficient theorem that
it (,C) ~ Ii (M,Z) ® C
Under the isomorphism H2(M,C) = r(M, C 2)/dr(M, de Rhem's theorem, the cohomology class
represented by a differential form
jp1)
c(g) E H2(M,C)
furnished by will be
sp(g) a r(M, F 2) ; and then
ffM Cp(g) e C will be the constant associated to that cohomology
class under the identification H2(M,C) = C introduced in Section 5. In fact, this will be an integer, and will be called the Chern class
of
g
and also denoted by
c(g)
.
A useful explicit form for the
Chern class in this sense is given as follows. Lemma 14.
(tCO) a Z1(V
Let
and suppose that
(ra)
defined in the open sets
,
0 *)
represent a line bundle
are nowhere-vanishing U(,
C"
functions
and satisfying ra(p) = r0(p)(g0a(p)12
for p e Ua fl U .
cp =tai as log ra E P(M, F2)
Then
defined differential form on M , and c(E) - !IM q) -
N-ti ffM as log ra
This is a straightforward matter of tracing through
Proof.
the identifications in the preceding paragraph, recalling the explicit form of the eoboundary homomorphism of an exact sheaf sequence as First, to pass from the line bundle
given in the proof of Theorem 1.
to its characteristic class, consider the exact sheaf sequence
0 --4- Z -- 0 !4 B* ---a. 0 . The cocycle (tCO) e Z1'Ut , will be the image under indeed, merely take
a
of a cochain
(aC43 ) E C1'(UZ
,
6,) ;
aa3 - 2ni log too , for any fixed choice of a
branch of the logarithm in each set Ua fl UO # 0 ; by suitably refining the covering, all such intersections can be taken to be simply connected, and the logarithms are thereby well-defined. characteristic class
c(g) a 1?(M,Z)
is represented by the 2-co-
cycle given by the coboundary of the 1-cochain
(aa3) , namely, the
2-cocycle cao, a Z2(Vt.,z) where c,,, = a07, - aar + 'CO
+ 'P7 + aya .
The
aCO
-
Now this 2-cocycle can be envisaged as belonging
Z2(Vt C) , and the homomorphisms given in de Rham's
to the group
theorem follow from the exact sheaf sequence
0
r0 d> e 1 d> e2
C
F e C f
1
as an element of
cap? = a
.
Introducing the subsheaf
of closed differential forms, consider the exact sequence
0- C -> G 0 d; 1-eochain
-_> 0
(ai
)
1 -> 0. The 2-cocycle
Z2('Ut
,
a ZI(l!'t ,
'a0 + a 9 + a' .
40 0)
(ca37) considered
will be the coboundary of some
0) ; the condition is that
Then
(da!) E Z1(VL, 91) . In fact,
referring to the first part of the proof, we shall merely take aL
= av , so that da' = 21 d log J
sequence
11 d
0 -- ° c
0 , the element (dam)
considered as an element of Zl(n , P 1) some 0-cochain do'00 = TP - Ta .
(Ta) a C0(V
,
Now from the exact
.
C43
will be the coboundary of
t'1) ; the condition is that
Then dTa = dTP , so
ferential form, a 1-cocycle of Z0(jk ,
selecting any C" differential forms
define a global dif-
(dTa) 4-
2)
.
To be explicit,
Ta of degree 1 in the sets
Ua such that
(2)
d log ga3 + TP
Ta = 2
The differential form
cp a r(M,
e2 )
in U a fl U . = dTa in Ua
defined by
is well-defined, and the Chern class (as an integer) is given by
C(g) - ffMdfP . Now, to finish the proof of the lemma, the functions ra are nowhere-vanishing, hence have well-defined logarithms; and these
logarithms satisfy
log ra(p) = log ri(p) + log gy(p) + log Since the functions and
gPa(p)
are holomorphic,
for p e Ua fl Up a log gPa = d log gP
a log spa = 0 ; thus the differential forms
clearly satisfy (2).
Ta =
log ra
The differential form p will then be given
by q)= dTa = 2n d a log ra - 2a a log ra ; and c(g) = tai fff as log ra
,
as desired. Remark.
There always exist functions
{ra(p))
having the
properties required in the preceding lemma, and indeed the functions
For introducing
can always be taken to have strictly positive values.
r
the sheaf
lection
of germs of positive-valued defines a 1-cocycle in
(Iga3I2)
desired functions
functions, the col-
C"
Z1(j
and the
* H1(M, , ) = 0 .
as its coboundary; thus it suffices to show that
' Nov the subsheaf W C (o
of real-valued functions is clearly fine,
and the ordinary exponential mapping
is a sheaf
exp:
H1'(M, .) = 0 , as desired.
*)
Hn-(M,
Theorem 11..
,
form a zero-cochain having this 1-cocycle
fra)
isomorphism; hence
2z,)
,
For any line bundle g e H1(M,
C9-*)
on a com-
pact Riemann surface M , and any non-trivial cross-section
f E r(M,'n2*(i)) , c(g) =
where V(f)
E V (f) P CM p
,
is the order of the cross-section
f
at the point
p e M , as defined in §4(b). Proof.
points
Since M
is compact there are only finitely many
p e M at which Vp(f) # 0 ; calling these points
pi , the
divisor of f has the form N (f) = Ei and the assertion of the theorem is that _ (Ua)
c(,j) = Ei Vi .
be a coordinate covering of M such that the bundle (gad) a Z'(2. ,
is represented by a cocycle
the covering is so chosen that each point pi hood
Vi
Let
for which
for some index
Vi C Ua
*) ; and suppose that
has an open neighborai
but
Vi fl Ua = 0
i
for a # ai .
The functions
ry *
f E r(M, "l
(g))
fa representing the cross-section
are meromorphic in Ua and satisfy fa = gapfP
in Ua fl UJ .
Ifal2
The functions
C" and nowhere-
are thus
I fa) l= I tM I2. I fO l 2 ;and
vanishing in va - (Uipi) fl Ua , and satisfy
these functions can be modified arbitrarily within the sets
without changing the functional equations. there are sets
It is thus evident that
C" , positive-valued functions % defined in the various
Ua , such that
ga= It
in UaflU,,
I2gp
in Ua - Ua n UiVi
ga = I fall
By Lemma 14, the Chern class of the bundle c(g) - 2n1 jjM as log Since
Vi
ga = IfaI2
41
is given by
g
= 2 jjM as log
gC'
on M - UiVi , and
as log ga = aa(log fa + log fa) = 0
since
fa
is holomorphic there,
it follows that Ei 11v
c(g)
as log so,
.
i
By Stokes' theorem, since
as log ga - d
log g. , secure that
1 c(j)
where
aVi
2i Ei
jaVi
is the boundary of Vi .
log ga
Now
gix = fa
c(g) = 2i T-i jaVi a log fa
2i i
on
aVi , so that
actually
=
javi d log fa
Ei Vi
by the residue theorem; this completes the proof. It is an immediate consequence of the preceding that all
meromorphic cross-sections of the bundle on the Riemann surface
g
have the same total order
M , where the total order is by definition
Vp(f)
e M p teristic
.
This can be taken as the definition of the charac-
class of a line bundle, since as we shall shortly see every
bundle does have a non-trivial meromorphic cross-section.
One further
useful trivial consequence of this theorem is the following. Corollary.
If
£*)
g a H1(M,
is a line bundle on the c(g) < 0 , then there are no
compact Riemann surface M such that
or equivalently,
non-trivial cross-sections of the sheaf m
r(M, tD (a)) = 0 . Proof.
then
(b)
P a
If
f e r(M, 0 (a))
and f
is not identically zero,
M vp(f) > 0 ; thus necessarily c(g) > 0
Again suppose that M
consider a line bundle
,
by the theorem.
is a compact Riemann surface, and
g e R (M, ® )
.
Introduce the expression
X(9) = dim RO(M, d (1)) - dim R1(M, S (9)) - cW ;
(3)
the cohomology groups are finite-dimensional complex vector spaces, and the dimension is meant in that sense.
Note that, applying the
Serre duality theorem, this expression can also be written in the form X(9) - dim P(M,
(3'.)
where
K
V (g))
- dim r(M, m (Kg-1)) - c(g)
is the canonical bundle of the surface M .
of the Riemann-Roch theorem is that this expression pendent of the choice of the line bundle proof of this assertion is the following.
$
.
,
The content X(9)
is inde-
A first step in the
Lemma 15.
Let n be a divisor on the compact Riemann H1(M,
surface, and let
n =
S*n9'
a
be the line bundle corre-
(¢ *)
sponding to that divisor, as in §4(b). I e n (M, m'*)
Then for any line bundle
,
x(gn) = x(g) . Proof.
Clearly it is sufficient to prove this assertion in
the case that the divisor is a single point, say 4 = 1-q .
Parallel-
ing the discussion in J4(b), introduce the subsheaf LI.,(,9 ,g) C 1rj(1)
defined by
)p = (f
f =? 0
J(f)>A near p);
or
since J- l q > 0, actually ( (A , E) C 9 (1) . sheaf 1 = (g (1)/ (DM(A , g) clearly has the form
p
0 1
The quotient
if p# q
C if p = q
(compare with the example discussed on pages 23 and 24).
As in
Lemma 7, it follows that
Or, (4 ,g)
mcgn)
.
There thus follows the exact sequence of sheaves
0 ---?
-->>
9 (g) ---. J -> 0
.
Consider then the associated exact cohomology sequence:
0 T H0(M, (9 (9i1)) - > H0(M, m (9)) - . ON, J) --. (4)
H'(M, ((gn))->H'-(M, m(g))-->Hl(M,-A)-->.-. . p
Since
J
is a skyscraper sheaf, having stalk
it follows readily that
C:
at a single point,
H°(M, J ) _ C and H''(M,I ) = 0 .
Now in
an exact sequence of complex vector spaces as in (4), the alter-
nating sum of the dimensions of the vector spaces is zero; this can
be rewritten as the equality dim HO(M, ) (gn)) - dim H-(M, Q! (gn)) + 1 (5)
= dim HO(M, m (g)) - dim Hl(M, (9(g)) Note that
(This peculiar observation results from the
c(n) _ -1 .
notational conventions adopted.
Recalling Lemma 7, and its preceding
discussion, the line bundle of a divisor 4 was defined to be the
element S*(A ) e H1(M, 9 *) (da)
in the exact sequence (2) of §4.
If
are local functions defining the divisor, then the functions
l/da are a meromorphic cross-section of the sheaf this convention.
rn*(,9 ) , by
The total order of the divisor n9
is thus the
negative of the total order of any meromorphic cross-section of its associated line bundle.
to the divisor
Hence, for the line bundle
it follows from Theorem 11 that
Then, replacing 1 in equation (5) by
c(W1)
, it follows that
associated c(n) = -1 .)
-c(n) = c(n-1) , adding
to both sides of the equation, and recalling that =
n
c(g)
c(g) + c(n-1) =
X(9n-1) = X(g) , which suffices to con-
clude the proof.
Now by using this lemma and the Serre duality theorem, it is an easy matter to prove the fundamental existence theorems on a Riemann surface.
The discussion in §4(b) should be recalled here.
Theorem 12.
On a compact Riemann surface M , H1(M,
'Y'*)
= 0;
equivalently, every line bundle on M has a non-trivial meromorphic cross-section, hence every line bundle is the bundle of a divisor.
Proof.
The equivalence of the three assertions of the
theorem was noted in §4(b).
Noting that a line bundle is the bundle
of a divisor on M precisely when the bundle admits a non-trivial meromorphic cross-section, it clearly suffices to show that, given any complex line bundle divisor such that
, there exists a line bundle
I
q
of a
has a non-trivial meromorphic cross-section.
11
In fact, we shall show more, namely, that given a line bundle
there exists a line bundle
of a divisor such that
rl
gq
bundle
r(M, Of (gi)) - 0
of a divisor on the surface M .
rj
q ; and since
for every line
(Kg-1n-1))
dim r(M, 61 (h )) a 0
- can) for all
q
by assumption, it follows that
dim r(M, m is independent of
rl
.
characteristic classes enough,
(Ki-In-1))
c(Kg_lq-1) = c(Kg-1) - c(q) < 0 .
is large; but then
pendent of
with arbitrarily given
rl
c(q) ; and by taking
to Theorem 11, it follows that c(q)
+ c(gn) - C
There are bundles
c(ry)
to be large
Thus, by the Corollary
dim r(M, V
c(1q) - c(g) + c(q)
q t which is absurd.
(KI'l1-1))
- 0 whenever
would also be inde-
This contradiction then proves the
theorem.
This existence theorem then shows that the study of divisors on the surface can indeed be reduced to the study of line bundles. In particular, referring to equation (3) of page 53, the divisor class group
A(M)
of the surf8'
M
-
By Lemma 15 the expression
x(gq) - dim r(M, 0 (fin)) - dim r(M, ® is independent of
has a non-
r(M, 6L(W) 10
trivial holomorphic cross-section, that is, such that Suppose, contrarily, that
g ,
is isomorphic to the group
H (M, 61) of line bundles on the surface. On a compact Riemann surface M
Corollary.
teristic
X(g) = dim H°(M, IV (9)) - dim H1'(M, 61(g))
the charac-
constant, independent of the choice of the line bundle Proof.
Since every line bundle
is a
- c(g) 9
.
on the surface M
g
is
the line bundle of a divisor as a consequence of the theorem, it follows from Lemma 15 that
X(g) - X(1)
for any
, which serves
to prove the desired result.
It is of course of some importance to determine the constant
for a given surface M .
X(g)
For this purpose set
1 , the
trivial bundle, and note (using Serre duality) that X(1) = dim r(M, (1) - dim r(M, (9l'0) - c(l)
= 1 - dim r(M, 6
1,0) .
The constant
g = dim r(M, S 1'°) ,
(6)
the dimension of the space of abelian differentials on the surface M , is caled the genus of the surface M .
This constant has a
simple topological interpretation as follows.
Considering the exact
sequence of sheaves
0 -? - B d>
m
1, 0
--> 0
the associated exact cohomology sequence has the form
0 - H°(M, C)
Hl'(M, ) --D H'1(M, since
?(M,
H°(M, 41, 0) ---> H1'(M, C)
H°(M,
110)
) = 0 by Theorem 8.
T
--> H2(M,cr) --- 0 , Now H°(M,C) = H°(M, ®) _ C ,
since all the global holomorphic functions on a compact Riemann surface are constant; and
H2(M,d") _
C
as noted earlier.
Therefore,
recalling that the alternating sum of the dimensions of the terms in a finite exact sequence of vector spaces is zero, it follows that
dim H0(M, 0
1,O)
(7)
- dim H1(M,C) + dim H1(M, m ) - dim HZ(M, Q 1,0) + 1 = 0 .
By definition,
dim H0(M, d
1,0)
= g , and by the Serre duality
theorem,
dim H1(M, 1D) = dim H0(M, 0
dim Hl(M, 19
110)
and
= g ,
1,0) = dim H0(M, &) = 1 ;
therefore (7) becomes
(8)
2g = dim H1(M,C)
That is to say,
H1(M,C)
is an even-dimensional complex vector space,
and its dimension is twice the genus of the surface M . X(g) = 1 - g .
The constant
As yet another interpretation of the genus, consider
the canonical bundle
K ; then
1- g = X(K) = dim H0(M, Q (K)) - dim H1(M, 0 (K)) - C(K) By definition,
dim H0(M, 0 (K)) = dim H0(M, ®l'0) = g ; and by the
Serre duality theorem,
=dim HO(M,IV )=1(9)
.
dim H1(M, m (K)) - dim H1(M, 0
1,0) =
Therefore
C(K) = 2(g- 1) ,
relating the genus to the characteristic class of the canonical bundle.
Since cross-sections of the canonical bundle are just abelian differentials, it follows from Theorem 11 that the total order of an abelian
differential on M is precisely 2(g -1)
.
In terms of the genus, the Corollary to Theorem 12 can be restated as follows. Theorem 13.
if M is a compact
(Riemann-Rock Theorem)
Riemann surface of genus
g
and
9 e Hl(M, 6 *)
is a complex line
bundle on M , then dim H°(M, 61(x))
dim H1(M, !
g ;
or equivalently,
dim r{M, &
dim r(M,
K
where
1 - g ,
is the canonical bundle.
In some cases, the Riemann-Roch Theorem furnishes explicitly the dimension of the space of holomorphic cross-sections of a complex line bundle; the following table may prove useful in keeping this in &7*)
As a notational convenience, for a line bundle
mind.
E e Hl(M,
we shall write
7(a) = dim r(M, 62(9))
(10)
Then:
(a)
cW < 0
=-> YW = 0
(b)
cW = 0
-= > 7(0 =
(c)
c(g) = 2g-2
=__>
(d)
cW > 2g-2 =s> 7(e) = c(9) - (g-1)
f
(]1)
7(9)
-
To see that '(11) holds, recall first that
1
if
=1
0
if
1
g
if
K
g-1
if
c(s) < 0
r(M, 01(9)) = 0 , by the Corollary to Theorem 11.
K
implies that
Furthermore, by
that same corollary, if
c(g) = 0 and
y(g) > 0
there must exist
at least one non-trivial (i.e., not identically vanishing) holo-
morphic cross-section of the line bundle have total order
c(g) = 0 ; that is, that cross-section must be
holomorphic and nowhere-vanishing on M . (cf. page 56), and hence
recalling that bundle
I , and its divisor must
7(g) = 1
c(K) = 2g - 2
.
This means that
Now when
g = 1
c(g) = 2g - 2,
and applying part (b) of (11) to the
Kt-1 , it follows that
7(Kf-1)
=
Then by the Riemann-Roch Theorem,
1
if
0
if
Kg-1
= 1
Kg-1
1
7(g) = y(Kg-1) + c(g) - (g-1)
= y(Kg-1) + g -I , from which (c) follows immediately.
c(g) > 2g- 2 , then
part (a) that
c(K9-1) = c(K) -
Finally, if
0 , so it follows from
7(Kg-1) = 0 ; and part (d) follows immediately from
the Riemann-Roch Theorem again. For the line bundles between the trivial bundle and the canonical bundle, that is, for those. bundles
I e H1(M, 0*)
such
that 0 < c(g) < 2g - 2 , the Riematin-Rosh Theorem merely provides
the equality y(g) = Y(K¢-1) + C(E) - (g- 1) ; and thus the formula merely relates two unknown quantities.
How-
ever, it is easy to qbtain some useful inequalities for line bundles in this range. (12)
First, since
7(Kg-1) > 0 , it follows that
y(g) > c(g) - (g -1)
for all
g e H1(M,
)
.
To obtain inequalities in the other direction, select a line bundle t
with cW = 1
and
7(t) > 1 ; for instance, the line bundle
associated to the divisor
can be taken as
5 , since it has
Now multiplying a cross-
at least one holomorphic cross-section h . section of any line bundle
by the section h yields a cross-
section of the line bundle
; and thus
for any index
y(gCr) > y(g)
this process,
y(gt) > y(g) r > 0 .
Repeating
.
if
c(g) < 2g - 2 , talk in particular r - 2g - 1 - c(g) ; then
C(gtr) = c(g) +
2g - 1 ,
so by (11d) it follows that Y(g5r) - g It then follows that
y(g) < g whenever C(Kg-1) < 2g
0 < c(g) < 2g - 2 , then
- 2
If
c(g) < 2g - 2 .
y(Kg-l) < g ;
so that
Y(g) = y(Kg-1) + c(g) - (g-1) <
and by the Riemann-Roch Theorem, c(g) + 1 .
-
Consequently, Y(g) _ min(g,c(g) + 1)
(13)
for
0 < c(g) < 2g -2 .
The following table may help to keep these inequalities in mind: (14)
c(g): 0 1
2g-1
2 ... g-2
g-1
g
g+1 ... 2g-3 2g-2
g
g
g
...
g
g
g
0
1
2
...
g-2
g-1
g
max Y(g): 1
2
3 ... g-1
min Y(g): 0
0
0 ...
0
We shall next see that for a general line bundle is the minimum given in the table.
g
the value
y(1)
(c).
At this stage it is possible to extend a bit further the
discussion begun in §4, concerning relations between divisors and
line bundles on a compact Riemann surface M select a complex line bundle section
f e r(M, (-(tp))
.
For any point p e M
tp e Hl(M, () , such that there is a
for which
J'(f) =
there always
exists such a bundle, since the line bundle associated to the divisor under the exact sequence (2) in §4 will do. 5P
In fact, the bundle
is determined uniquely by the point p ; for if
other such bundle, with then
f'/f e 1(M, (¢(
f' a r(M, -()) its corresponding section, /tp))
section, hence necessarily
is a holomorphic nowhere-vanishing !/gyp = 1 .
Line bundles of this form
will be called point bundles on the surface M . for any point bundle
were any
Note that
5p , by Theorem 11; and that
c(sp)= 1
y(ip) > 1 , since
by definition a point bundle has at least one non-trivial section.
If M is a compact Riemann surface of genus
Lemma 16.
g > 0
and
is a point bundle on M , then
5p
y(ip) = 2
by the inequalities (14).
independent sections
M -- ;;- 1-1
2 ; of course,
Suppose contrariwise that
Proof. then,
y(ip) = i .
Select two linearly
fl,f2 e r(M, 9-(tp)) , and consider the mapping
defined by q e M --4 (f1(q), f2(q)) a
(15)
P
,
in terms of homogeneous coordinates on the projective line.
the values of
fl(q), f2(q)
Since
in two different coordinate neighborhoods
are multiplied by the-same non-zero constant when passing from one neighborhood to the other, these values determine the same point in the projective line.
If both functions
fl, f2
vanish at a point,
then dividing by their greatest common divisor will yield the same mapping into projective space, thus extending that mapping to the common zero of the functions
fl, f2 .
It is thus evident that (15)
is a well-defined complex analytic mapping from M into For any point
(al,a2) e P1
consider the section
By Theorem 11 again, there is a unique
e2fl - alf2 a r(M, o (s))) .
point
N'1
q e M for which this section vanishes, that is, for which
(fl(q), fq(q)) = (a1,a2) e
The mapping (15) is therefore one-
.
to-one, so that the surface M
is analytically equivalent to the
, and hence the genus must be
surface
This contradicts
g = 0 .
the hypothesis, and therefore concludes the proof.
Now if M has genus bundles equal.
are equal if and only if the points
gyp, Sq
For if
g > 0 , it follows that the point
p # q , then the bundle
for
tp = tq
are
p, q
tp = tq
would have at least two non-trivial sections, one vanishing at and one at
q ; and thus
p T 5p
y(ip) > 2 , which is impossible by the
Therefore, if M has genus
preceding Lemma.
p
g > 0 , the mapping
is a one-to-one mapping from the surface M to the subset
of H1(M, (9-*)
consisting of those complex line bundles
which cQ) =
1 .
For the case of genus
the Riemann-Roch theorem that cQ) = 1 thus as in the preceding Lemma,
M = 1P
0 ,
implies .
C
for
it follows from
yQ) = 2 , and P1 there exists
On
a
meromorphic function with an arbitrarily prescribed simple zero and simple pole; so any two points and thus
Sp = tq
unique line bundle
p,q
for all points
on
1 IF
are linearly equivalent divisors,
p,q e P1
.
There is hence a
with cQ) = 1 .
That is, if M
has genus
g = 0 , then M = P' and all the point bundles of M
coincide.
An arbitrary complex line bundle on M can be built up from these point divisors.
for which
First, if
g e H1(M, (*)
is a line bundle
y(g) > 0 , select a non-trivial section f e r(M, 19-(g))
and write r(f) _=1 1-pi , where pi e M need not be distinct.
r = c(g)
and the points
_ p ..
It is then clear that
;
r
i
for if
fi a r(M, 6-(Sp ))
are non-trivial sections of the point
i
bundles, so that
fi
e r(M,
f1fl...
vanishes precisely at pi , then is a holomorphic, nowhere-vanishing
(g5-1. ..C,p1))
i
pl
section, so necessarily g1... -1 = 1 .
This representation of
pr
course only holds for bundles it follows from (i4) that possible.
g
with
7(e) > 1 ; but if
c(g) > g
y(g) > 1 , and such a representation is
Next, , for a general line bundle
g e H1(M, 6L*) , let
and select some point p e M as base point; then
r = c(g)
g , so that as above there is a representation
,,-r
It therefore follows that, having selected a r base point p e M , an arbitrary bundle g e H1(M, 9 ) can be .
represented in the form where
pl...,pg r-g
(16)
the points
r = c(g)
pi e M depending upon the bundle
The representation (16) may not be unique; if it is not, there will be points
gl,...,gg e M such that
r-g = Cpl...5pgfip
Sr-8
...C
C
ql
sets
hence such that 9g P
(pl,...pg}
p
t
g
01,...gg} .
= 5
, where the
...5
gl
q8
This then means that 7Qp ...Sp ) > 2 1
r
and another with divisor
for there is one section with divisor
Since the converse is clear, it follows that the representation
(16) is unique precisely when yQ
...C
Pg
P1
) a 1
.
To examine this condition further, consider more generally a complex line bundle
; the associated divisor AO = 1-Pi Pr , where the qi are the distinct points
..
f -
p1
can be written AO
a1 vigi
occurring among the points pl,...,pr .
Let
hl,...,hg E r(M,O-(K)) be
a basis for the space of Abelian differential forms on the surface M ,
and as an abbreviation let b
denote the column vector
4 h =
h In terms of a coordinate system functions
hi
) zi
centered at the point
are complex analytic functions of the complex variable
in an open neighborhood of the origin.
zi
qi , the
The values of these func-
tions and of their derivatives at the origin are well-defined, and $ and the corre-
will be denoted by hj (gi), hl,(gi),...,h j
sponding column vectors will be denoted by h(gi),h'(gi),
" ''h(v(gi), ..In
terms of a different coordinate system centered at the point a different set of values will of course be obtained; the vector
qi
h(qi)
will be replaced by a nonzero constant multiple of itself, and a vector h(v)(gi)
vectors
(17)
will be replaced by a suitable linear combination of the h(gi)t h'(gi),...,h(v)(g
Thus the rank
p = rank(h(g1),h'(g1),...,h(vi
at least will be invariantly defined; the matrix in (17) has
1)(ql);...;h(ga),h'(gs)'...,h(vs-1)(qs)
r = v1+...+vs
Lemma 17.
For a complex line bundle
...
J _
pl
pact Riemann surface of genus
on a comr
g ,
Y(9) - r -P+l where
is the rank of the matrix (17).
p
Proof.
Since, c(l) = r , it follows from the Riemann-Roch
theorem that y(e) = r- g+1+y(Kg-1) the vector space
.
fl,...,ft
If
r(M, 6-(Kg-1)) , where
is a basis for
t = y(Kg-1) , and if
gi a r(M, 0-(sp )) , so that N9-(gi) = 1-pi , then clearly the elements i
r(M, ti(K)) ,
figl " .gr a
1 < i < t , form a basis for the subspace
of Abelian differentials consisting of those elements
(h) > 90 , where
such that
^¢ 0
Ei=1
l-pi .
y(Kf-1) = aim(h a r(M, & (K))
h e r(M,
Thus
p9(h) > o)
Letting h1,...hg be a basis for the apace of Abelian differentials, h E r(M, @-(K))
any element
h = c1h1 + ... + cghg
can be written uniquely in the form
for some complex constants
ci
The condition
.
A0 that 4 (q)) >
just means that h(qi)
= EE cjhj(gi)
=0,
ht (gi)
= EE cJhi(gi)
=0,
h(Vi-1)
(gi) = Ej cihj(vi-l) (qi) = 0
where
qi
and this in
are the distinct points in the divisor
turn means that the row vector matrix (17).
(cl,...,cg)
is annihilated by the
Consequently y(Ki-l) = g - p , where
p
is the rank of
the matrix (17); so that y(t) = r - g+l + y(Kt-1) = r - p + 1 , as desired.
We dhall return later to exploit this result more thoroughly, but for the present shall be content with some simple observations.
When
r = g , the matrix appearing in (17) is a
g X g
square matrix;
the vanishing of its determinant is equivalent to the condition that p < g , and hence by Lemma 17 is also equivalent to the condition that
y(1) > 1
Therefore
.
y(j) = y{ (v
(18)
det(h(g1),h'(g1),...,h
where 7=1 Pp i
v1q.3.
i
(v -1)
(q1);...;h (gs),h'(ga),...,h
and the points
pi
qi
are distinct,
det(h(p1),h(p2),...,h(p9)) = 0
posing the vector q
if and only if
pg
i)
1
particular, when all the points if and only if
) > 1
...t
p1
.
s
(qa)
are all distinct.
In
y(Sp ...tp ) > 1 The functions com-
are linearly independent, so that it is evident
that this determinant does not vanish identically; there are thus always distinct points
pi
pi
so that
are distinct points of M and
y(C
Ui
about them with coordinate mappings det(h(zl)h(z2),...h(zg))
of g
z
:
i
) = 1 Indeed, if pg are coordinate neighborhoods p1... C
.
Ui ---> C , then the function
is a non-trivial complex analytic function
complex variables in the domain
U1 x ... X U9 C Cg ; the set
of points at which this determinant vanishes is a proper analytic sub-
variety of U1 X ... X U9 , so that in this sense for a general set of
g
distinct points
y(tp ...gyp ) = 1 1 g
pl,...,pg .
Thus in general
the representation (16) is unique; and in the same sense, as the reader will easily verify,
y(g)
in general takes the minimum value in
table (14). Note in passing that if
hl,...,hg a r(M, (Q (K))
is a basis
for the space of Abelian differentials on a compact Riemann surface M of genus
g > 0 , then the functions
hi have no common zeros on the
surface M
.
For if p is any point of M,
y(cp) = 1 by Lemma 16;
then applying Lemma 17 in the particular case that
1 =
so that
(a)
2 - p , hence
that
p = 1
r = 1 , it follows
where
p
rank(h(p))
,
for at least one function hi .
hi(p) # 0
Of particular interest are divisors of the form
equivalently line bundles of the form
point on the surface and
V = 1,2,3,...
always an Abelian differential
p .
p e M is a given
, where
As noted above, there is
hl c r(M, (Q (K))
at the point p , so that p(hl) = 0 .
or
Let
which is non-zero
h2 a r(M, 6L(K))
be
an Abelian differential which vanishes at p , but such that
p(h2) h3 e r(M,
P2 - 1
is the minimum possible value.
be an Abelian differential which vanishes et p
9 (K))
least to the order mum possible value. hl.,h2,...,h9
Then let
p2 , but such that
V(h3) = p3 - 1
at
is the mini-
Continuing in this manner leads to a basis
for the space of Abelian differentials on the surface
M , such that P(hi) = Pi -1 where
=P1
(19)
(Since the total order of an Abelian differential on the surface is 2g - 2 , as noted above in discussing equation (9), it follows that
p(hg)
pg- 1 < 2g-2 , hence that pg < 2g .)
this sequence of integers
pi = pi(p)
It is clear that
is uniquely determined by the
point p c M ; it ism1led the Weierstrass gap sequence at the point p , and the values
the point p .
pi = pi(p)
are called the Weierstrass gaps at
The significance of this sequence is indicated by
the following result.
Theorem 14.
point p (a)
Consider the Weierstrass gap sequence at a
on a compact Riemann surface.
Then
y( V) = v+ 1 - (number of gaps < v) ; so that if
1
otherwise.
V
is a gap,
-
p (b)
0 y(tv-1)
YaV)
p
There exists a meromorphic function on the surface, whose only
singularity is a pole of order precisely and only if
V
Proof.
V
at the point
p , if
is not a gap.
Let h denote the column vector consisting of a
basis for the Abelian differentials on the surface, where the basis is normalized as above so that gap sequence
at p .
(pi)
vp(hi) = pi - 1
for the Weierstrass
Then it is evident that the matrix
has the form
(h(p),h'(p),h"(p ),...)
1
2 ... p2
0
0 ... *
0
0 ... 0
p2+1 ... p3
p3+1 ...
(20) ... ...
...
0 ...
... * ...
where the headings denote the numbers of the columns, and * for a non-zero constant.
sisting of the-first p = (number of gaps
Y(Sp)
v
For an integer
stands
V > 1 , the matrix con-
columns of (20) clearly has rank
pi < V)
.
Therefore from Lemma 17 it follows that
= V- p+l = v+1 - (number of gaps pi < v)
The second part of assertion (a) is an obvious consequence of this formula.
As for assertion (b), let section, so that ,9-(g) = f/gV
g e r(M, (.(gyp)) be any non-trivial
Then if
f e I'(M, 6Qp)) , the quotient
will be a meromorphic function on the surface M , whose only
singularity is a pole at
p
of order at most
meromorphic function can be so realized.
V ; conversely, any such
Therefore
y( V)
is the
dimension of the space of meromorphic functions on M , whose only singularities are poles at gap, then
p
of orders at most
V .
is not a
y(( pV) = y(V-1)+ 1 , so there must exist a meromorphic
function on M whose only singularity is a pole at
V but not at most
the order is at most order is precisely y(5pV)
If V
= yQpV-1)
V .
p , and such that
v -1 ; in other words, the
On the other hand, if
v
is a gap, then
; so for any function meromorphic on M whose only
singularity is a pole at
p , if the order of the pole is at most
then it must actually be at most
V ,
This serves to complete the
V - 1 .
proof.
For any given.point p e M the Weierstrass gaps at the point p
pi(p) > i
satisfy the inequalities
trivial consequence of (19).
for
i = 1,2,...,g , as a
The discussions in the preceding part (c)
might lead one to suspect that in general the Weierstrass gaps would actually take on these minimal values; for by Theorem 14 that would correspond to the dimensions
y(Sp)
being least possible.
(Since
y(5P) > 1 , the least possible values for these dimensions differ from the general results given in table (l4) in that the zeros of that table are replaced by ones.) are possible.
This is indeed so, and quite precise results
The Weierstrass weight of a point p e M is defined in
terms of the Weierstrass gaps
pi(p)
at
g (21)
w(p) = Z (Pi(p)- i) i-1
p by
Note that
w(p) > 0 for all points
called a Weierstrass point if
A point p e M
p e M .
w(p) > 0
.
is
The Weierstrass points
on the surface M are thus those points at which the Weierstrass There are actually
gaps do not attain their least possible values.
only a finite number of Weierstrass points on any Riemann surface, as a trivial consequence of the following result.
If M is a compact Riemann surface of genus
Theorem 15.
g , the Weierstrass weights of the points on the surface M satisfy the equality
E
peM Proof.
Let
w(p) = (g-1) g(g + 1)
h1,...,hg a r(M, Q(K))
again be a basis for
the space of Abelian differentials on M , and let column vector of length
formed from this basis.
g
h
denote the If
(Uaza)
is
a coordinate covering for the Riemann surface M , then on each set za(Ua) C C this vector is a column of complex analytic functions h(za) ; and for points in Ua n u, , where the coordinate transition function is
za = fa$(z0) , these column vectors satisfy where
h(za) = KC43 h(z
equation with respect to
Koo(zo) = dz /dza
.
Differentiating this
za , it follows that
h'(za) _ (dz/dza) dz0
h(zp)) = K2a (z.) h'(zp) + (*) h(z,)
where (' ') stands for some holomorphic function.
Continuing in this
manner, in general (22)
h(v)(za)
=
KV+l(z0)h(v)(z
+ (*)h(v'1)(z
+ ... +
The function ga(za) =
det(h(za),h'(za),...,h(g-1)(za))
,
is then holomorphic in
za(Ua) ; and from (22) it is evident that
in UafUJ,
-
Kap(zn)1'+2+...+(B-1).g,3(zp)
ga(zes) -
)g(g-1)/2,5(z
z
That is to say, the functions Kg(g-1)/2
line bundle
E
g = (ga(zes))
AL-1)
V (g) =
peMP
The order
vp(g)
define a section of the
Then, by Theorem 11 it follows that
.
(23)
)
c(K)
2
_ (g-1)6(8+1)
is of course unchanged when the functions
are
hi
subject to any nonsingular linear transformation; so when considering a point
p e M there is no loss of generality in supposing that
vp(hi) = pi(p)- 1 , where If
z
(pi(p))
are the Weierstrass gaps at
p
is a local coordinate mapping defined in a neighborhood of p
and such that of the variable
z(p) = 0 , then in the power series expansion in terms
z , the function hi(z)
order precisely pi- 1 .
will begin with a term of
The lowest order terms in the power series
expansion of the function
g(z)
then obviously come from the
expansion
(p1_l)z zp2-1
P 1-g\
p 1-2
P 1-1 z
... (p1-1)...(pi g+l)z
(P2-1)zp2-2
(p2-1)...(p2-g+l)zP2-g
...
det
P -1 z g
p -2 (Pg 1)z g
P -g (pg 1)...(Pg-g+l)z g
p-1 i Since the functions
z
are linearly independent, this Wronskian
determinant cannot vanish identically; and since each monomial in
the expansion of the determinant has order (P1+P2+...+Pg) - (1+2+...+g) _ (p1-1) + (p2_2)+...+(Pg-g) = W(P) that is the order of the full determinant.
That is to say,
vp(g) = w(p) ; and upon substituting this into equation (23), the desired result follows immediately, thus concluding the proof. According to Theorem 15, a surface of genus no Weierstrass points. while for genus
0
or
has
1
This is of course trivial for genus
0 ;
1 , it could have been noted as a consequence of
the last remark in part (a) above, since in that case the unique Abelian differential on the surface is nowhere zero. quence, the canonical bundle of a surface of genus the trivial bundle.)
(As a conse1
is necessarily
In general, it is clear that the minimum pos-
sible weight for a Weierstrass point is
w(p) = l ; and that this
corresponds to the Weierstrass gap sequence of the form 1,2,...,g-l,g+l, where
g
is the genus of the surface.
A Weierstrass point is called
a normal Weierstrass point if it has this minimal form, that is, if
w(p) = 1 ; and the surface M is called a normal Riemann surface if all of its Weierstrass points are normal Weierstrass points. Theorem 15, a normal Riemann surface has precisely Weierstrass points.
By Theorem 14, the dimensions
By
(g-1)g(g+l) 7( V)
read off immediately, and depend merely upon whether or not
can be p
is
one of the Weierstrass points; and the meromorphic functions having but a single pole on the surface must have a pole or order at least g . The consideration of the other extreme behavior of Weier-
strass points is a bit more subtle, and depends on the following observation.
For a fixed point p
on the surface, suppose that
are non-gap values; then by Theorem 14 there are meromorphic
v1, V2
functions
f1 f2
on the surface, such that
singularity a pole of order precisely product
vi
fi
has as its sole
at the point p .
has a pole of order precisely V1 + V2
f1f2
is regular otherwise; so by Theorem 14 again, non-gap value.
V1 + V2
at
The
p , and
is also a
Therefore, the set of non-gap values at a point is
closed under addition, (forms an additive sub-semigroup of the positive integers).
point V- r
Letting r
be the least non-gap value at the
p , it follows that whenever
V > r
is a gap value, then
is also a gap; consequently, all the gaps occur in finite i, i+r, i+2r,...,i + X r , (where
arithmetical sequences of the form i = 1,2,...,r-1
i
and X i
In particular, a point p
is called a hyperelliptic Weierstrass point if its least non-gap value is
form
2 ; at such a point the Weierstrass gap sequence has the The weight of a hyperelliptic Weierstrass
point is
w(P) _ [1 + 3 + ... + (2g-1)] - [1 + 2 + ... + g]
= [1+2+... +2g] - 3[1+2+ ... +g] =2g(g-1) A Riemann surface is called a hyperelliptic surface if all of its Weierstrass points are hyperelliptic Weierstrass points; by Theorem 15, such a surface will have Theorem 14, the dimensions
2(g+l)
y(V)
depend merely upon whether or not
Weierstrass points altogether.
By
can be read off immediately, and p
is one of the Weierstrass
points; and for each hyperelliptic Weierstrass point there exists a meromorphic function having a double pole at that point but being
regular otherwise.
surface of genus
an elliptic Riemann surface.
g = 1
is called
For such a surface, it follows from
the Riemann-Roch theorem that
y(c2p)
therefore, for every point
on an elliptic surface there exists
p
= 2
for every point
p ; and
a meromorphic function having a double pole at that point but being
For this reason, elliptic curves are sometimes
regular otherwise.
considered as falling within the class of hyperelliptic curves, even though they have no Weierstrass points at all.
However we shall not
adopt-this convention, but shall distinguish between elliptic and.
hyperelliptic surfaces; so hyperelliptic surfaces all have genus
g>1.) Now turning to the case of a general Weierstrass point, the following assertion can be made. Theorem 16.
If p
is a Weierstrass point on a compact g , its weight satisfies the inequality
Riemann surface of genus
1<w(P) <2g(g-1) ; w(p) = 1 precisely when p
precisely when p
w(p) = 1 g (g - 1)
point.
is a normal Weierstrass point, and is a hyperelliptic Weierstrass
The total number N of Weierstrass points on the surface
satisfies the inequality
2(g+l) < N < (g-1) g(g+l) ; N = 2(g + 1)
precisely when the surface is hyperelliptic, and
N = (g - 1) g(g + 1)
Proof.
precisely when the surface is normal.
Letting
Weierstrass point
r > 1
be the least non-gap value at a
p , the gap values can be written in the form
i, i+r, i+2r,...,i+Xir , (i = 1,2,...,r-1; Xi = 0,1,2,...)
.
The total number of gaps is
r-1
r-1
i=1
i=1
(24) The weight of the point
p
is then
r-1 E
W(P) =
E (i+Jr) - 2 g(g+1)
i=1 J=0
r-1 =
E [i(xi+l) + 2rXi(,`i+l)J - 2g(g+l)
i=1
x
r (r-1) + E lXi(2i+rxi+r) -2g(g+l) i=1
Since
(i+ rXi)
is a gap value, it follows from the inequality (19)
that (i+r.i) < 2g-1 ; so 21+r.i+r < i+(2g-1)+r < (r- 1) + (2g - 1) + r = 2(g+ r - 1)
.
Then from (24),
rE 2(21+rxi+r) < (g+r-1) i=1
rE i=1
i=
(g+r-1)(g-x+1)
Therefore
w(p) <2g(g-1) - 2(r-1)(r-2) The maximum value of attained only for
w(p)
is therefore
g(g -1) , and this is
2 r = 2 , which is the hyperelliptic case.
assertion about the minimum value for
w(p)
The
was proved earlier.
Finally, the results about the total number of Weierstrass points follow immediately from the inequalities on the weights and Theorem 15, thus concluding the proof. For emphasis, recall again that surfaces of genus 1
have no Weierstrass points.
0
or
.
§8.
Picard and Jacobi varieties
(a)
The first step in the detailed classification of complex
line bundles over a compact Riemann surface M was taken in §7(a), with the introduction of the Chern class of a line bundle; from the exact sequence of sheaves
0 ---> Z -- > 9 e-> CT* --> 0
(1)
there followed the exact cohomology sequence
0 -> Hl(M, d )/H1(M,Z) -o> Hl(M, d)*) -t-> Z --> 0 ,
(2) where
is the homomorphism associating to a complex line bundle
c
its Chern class.
The subgroup of complex line bundles having Chern
class zero is thus isomorphic to the group
H3(M, (4-)/H1(M,Z) , and
the investigation of this group is the next step.
To begin, consider
the following exact sequence of sheaves
0
(3)
where
d
--_> C --. B- a > C} 1, 0 __> 0 ,
is the operation of exterior differentiation.
The associated
exact cohomology sequence over the Riemann surface M begins
c
0 - r(M,C) T r(M, - HH(M,
H1(M,
Since M is compact, Furthermore, H1(M,
1'0)
r(M,
L4l'o) S-> H-(M,C) --->
H2(M,C) --> H2(M, & ) -->
...
r(M,C) = r(M, C9) = C , and also H2(M,C)
=.D
H2(M, 69) = 0 by Theorem 8, and HO(M, C-) = C by the Serre duality theorem.
Therefore
the above exact cohomology sequence leads to the exact sequence
0Tr(M, 011'°)-5-> iI'(M,c)-->Hl(M, CO-) ---> 0.
(4)
Since the inclusion Z C t can be factored through the inclusions
Z C C C 9 , it follows that the homomorphism H1(M,Z) ---> H1(M, ( - ) in the exact sequence (1) can be factored through the homomorphism
H1(M,C) -4 H1(M, (9) in the exact sequence (4).
Hl M 0 ) ti
(5)
Hl(M,C)
H1(M,Z) + SP(, (Q1,0)
(M, z)
where
is the coboundary homomorphism arising from the exact
6
sequence (3).
For later purposes, recall that this homomorphism has For any element
the following explicit form.
cp a r(M, m 1'0) , and
for any suitable coordinate covering 7Z = (Ua) exist holomorphic functions Ua .
fa a r(Ua, (Q-)
of M , there will
such that
cp = dfa
in
The constants (SCP)
form a one-cocycle NO class
Consequently
5q) a H1(M,C)
.
e
= f0 - fa
Z1(2
C) , representing the cohomology
Note further that this homomorphism is actually
an injection; indeed, the following somewhat stronger assertion can be made. Lemma 18.
Consider the homomorphism
for a compact Riemann surface M .
If
cp a r(M,
such that
By a H1(M,M) C HI(M,C) ,
then 9 = 0 .
S: r(M, p 1'0)
110) --- H1(M,C)
is an element
Proof.
morphism
In terms of the above explicit form for the homo-
S , it is apparent that
for a suitable coordinate covering VC= (Ua) there exist holomorphic functions
in Ua and such that
f0 - fa
if and only if,
Scp a H1(M,ffi)
of the surface M ,
fa a P(Ua, d1-)
The functions
is real.
Igaj _ IgJ
ga = exp 2711 f a are thus holomorphic in Ua , and Ua n u, .
T = dfa
such that
Since M is compact, the globally defined function
must attain its maximum at some point of M ; but then
g
s
in IgaI
is con-
stant in an open neighborhood of that point by the maximum modulus theorem, and hence all the functions
The functions
theorem for analytic functions. constant as well, so
are constant by the identity
ga
are necessarily
fa
cp - dfa = 0 , as asserted.
Another, although equivalent, approach to the classification can be made through a slightly different exact sequence of sheaves. For any germ of function
f e
(9
define
d2(f)=211 dlog'f this is clearly a sheaf homomorphism dE
a
0:l'0 0l'0
d2:
.
is onto; for any germ of holomorphic differential form
can be written
cp = dg
for some function
g e
The mapping cp e
0-, and then
is the subsheaf CC (-
cp = d8(exp 2711 g) .
The kernel of
constant functions.
There is thus the exect sequence of sheaves
d2
X1,0
0 ___> C --? m * U > fl 1' 0 --? 0
.
The associated exact cohomology sequence over the Riemann surface
M begins
of
,
T Hl(M,
_
_ '-, Y
*) 2 Hl(M, 6'1'0)
(M,C) = C
.
Furthermore,
Al
,1'1, M. / ---sa
H2(M,C) --> H2(M, m*)
Since M is compact, r(,C*) = r(M, H2
/
6L*) _
...
C* , and also
H2(M, tD-*) = 0 ; for from the exact
cohomologr sequence associated to (1) there is the segment
H2(M, L, ) --> H2(M,
*)
T
Theorem 8, and H3(M,Z) = 0 duality theorem shows that c*
image of the mapping
C
H3(M,Z)
, while H2(M, V) = 0 by
since dimension
(M) = 2
The Serre
.
H1(M, 61,0) = HO(M, @ ) = C ; and the is hence the kernel of a homomorphism
ti
(Note that Z is the kernel
and will be denoted by Z . *
of a particular homomorphism C -? C
as might be expected, we
;
N shall shortly identify Z with Z .) As a consequence of these observations, the above exact cohomology sequence leads to the exact sequence
(6)
o -> r(M, 6 110) s Hl(M'C)
R1 (M' 0' *) -L> Z
0
The group H1(M,C) will be called the group of flat complex line
bundles over M .
The homomorphism
consists of those complex line bundles cocycles
is that induced by the
i*
* natural inclusion mapping C --k 01
;
so the image of g
i*
thus
which admit representative
consisting of constant functions. The Serre duality theorem merely asserts that
is canonically dual to the group
morphism H1(M, 19
110)
HO(M, CL ) - C ; the actual iso-
= C therefore involves an element of choice,
which will be made in the following manner.
class / a H1(M, a 1
H1(M, QL1,0
'O)
Considering a cohomology
as a linear functional on C , associate to
.
mat, Uonomo.i.ogy class zne complex Constant 7t-1) choice, the homomorphism c*
WLZJI tnls
.
in the exact sequence (6) can be
described as follows.
Lemma 19.
For a complex line bundle
a compact Riemann surface M ,
9 e h1(M, B *)
over
c*(9) E C is the Chern class of
that bundle.
This is a straightforward matter of tracing through
Proof.
First, the explicit form of the
the various mappings involved.
duality in Serre's theorem was described in §5(b). mology class
a e H1(M, X1'0)
(a01) a Z1( UZ ,
select a representative cocycle
in terms of a suitable locally finite coor-
ffl 1'0)
dinate covering
of M .
_ (Ua)
There is a zero cochain
(Ta) a C0(.A , t 1,0)
having coboundary
in Ua n U .
'r
Then
(aTa) a r(M, 1e1")
For any coho-
(a01) , so
a01 = T13 - Ta
= aTa , and the differential form
represents the cohomologyr class
Dolbeault isomorphism Hl(M,
61110)'_-'Y r(M,
a
01'1)/6r(M, e 0,1)
Thus the constant corresponding to the cohomology class our chosen isomorphism H1(M, ig 1,0) =11C
Next, for a complex line bundle cocycle
a
Q01) a Z1('In , 0*) , the image
by the cocycle
aa1 =
d.e(t
)
C43
fiRr
r1 =
raIt01I2
in the
*)
a
.
under
- IIM(?Ta) . with a representative
a = c*Q)
d log tat .
select nowhere-vanishing e functions so that
is just
H1(M,
under the
is represented
As in Lemma 14,
ra in the various sets
u.-fl U1 ; then
1101a1ogg 01=27-1i7 (a log rl-a logra) Thus in the explicit form of the Serre duality mapping we can put
-133-
Ua
ffM 2fiI aa(log ra) = 2.1ifi fff as log(s)
c*(9)
However, by Lemma 14 the latter integral is precisely the Chern class
c(O
, concluding the proof. It follows immediately from Lemma 19 and the exact sequence
(6) that the complex line bundles arising from flat line bundles are precisely those complex line bundles of Chern class zero; or equivalently, the necessary and sufficient condition that a complex line
bundle
g e H1(M, (9
)
admit a representative cocycle Q a$)
sisting entirely of constant functions is that
con-
c(L) = 0
In summary of the preceding, the group of complex line bundles of Chern class zero can be described in the following three equivalent forms:
H1(M, C)
HZ(M,z)
(7)
+. br(M, 1 0)
(9 EH-(M, O)Ic(0=
Moreover, the isomorphisms (7) lead to isomorphisms between the three groups on the right-hand side, which are explicitly as follows. First, from the exact cohomology sequence associated to the exact sheaf sequence (1+) there arises the homomorphism
H1(M,C) --k Hl(MO
which induces an isomorphism
H1(M,C)
H1(M,z) + Br(M, 41'0)
-134-
--? H1(M,(.) H1(M,Z)
Next, from the exact cohomology sequence associated to the exact sheaf sequence
0-Z-> C -kC*->0 there arises the homomorphism H1(M,C) T H1(M,C) , which induces an isomorphism
H1 M,C )
(M, C)
H1(M,Z)+ sr(M, l'0)
s*r(M, m1'0)
The first assertion was proved above; verification of the second assertion will be left to the reader.
(b)
The expressions (7) permit an additional structure to be
imposed on the group of complex line bundles of Chern class zero.
in general, a lattice subgroup of a finite dimensional real vector space is defined to be an additive subgroup of the vector space generated by a set of elements which are linearly independent over the real numbers; and a lattice subgroup of a finite dimensional complex vector space is defined to be a lattice subgroup of the associated real vector space.
Thus in an n dimensional complex
vector space, with its natural associated structure as a 2n sional real vector space, a lattice subgroup has at most generators.
H1(M,C)
dimen-
2n
Considering the first form given in (7), recall that
has the natural structure of a 2g
vector space, where
g
dimensional complex
is the genus of the Riemann surface
M .
As is known (cf. the topological appendix), H1(M,Z) C H1(M,C) is a lattice subgroup; indeed, any 2g a basis for the complex vector space
generators of H1(M,Z) H1(M,C)
.
form
By Lemma 18 the
_-
.. r....
image is a
. o.,, g
/ - n Mu) is injective, hence the
dimensional complex linear subspace; so the quotient
apace V - H1(M,c)/5r(M, 0-110)
has the structure of a XI,...
g dimensional complex vector space.
Let
? g be a set of generators for the lattice subgroup
Hl(M,Z) C H1(M,C) ; and let 1,...'X2g be the corresponding elements it the quotient space
lattice subgroup of V . not all zero, such that Ei x X
i i
a 5r(M,
imply that
V .
Then
generate a
Tel,...
For if there were real numbers
x1,.... x2
,
g
Ei xi i 0 , then
0-1'0) C H1(M,C) ; but then by lemma 18 this would
Ei x X
i i
= 0 , contradicting the fact that
generate a lattice subgroup.
Tl,...,T2
As a consequence, the quotient group
.El(M'CC)
H (M,z)+Sr(M, 61-1,0) has the structure of the quotient space of a vector space
g dimensional complex
V by a lattice subgroup generated by 2g elements;
this structure will be called the Picard variety of the Riemaan
surface M , and will be denoted by P(M)
v = H1(M,c)/Br(M, (11,0)
.
As remarked above,
Hl(M, (9 ) , so the same structure can be
described in the form P(M) = H1(M, O-)/H1(M,Z)
.
The Picard variety
is in particular an Abelian group, and its role in the classification of complex line bundles lies in its occurrence in the exact sequence
(8)
0 ---> P(M) --+ H1(M, (9*) -c
--40. 0
The group of all complex line bundles over M therefore has the natural structure
H1(M, (S*) ti Z + P(M)
To consider in more detail this additional structure, let
V be any g
dimensional complex vector space, and
lattice subgroup of V .
a° C V be a
Of course, as an abstract group
for some integer r < 2g , which will be called the rank of the lattice subgroup;
the rank clearly can be characterized also as the
dimension of the real vector subspace of V spanned by the elements
of it
.
We shall consider here only the case in which X has the
maximal possible rank.
sider V as a 2g of
V/Z
First, ignoring the complex structure, conA set of generators
dimensional real vector space.
can be used for a basis for the real vector space V , so that
lpg/Z2g
=
The space
(B/Z)2g .
as a Cartesian product of 2g
V/ a"e
can thus be factored
circles, and thereby has the structure
of a compact manifold of dimension 2g .
The vector space
V
is
obviously the universal covering space of this manifold, the covering
mapping being the natural projection V - V/o° 7rl(V/ K ) _ Z2g .
; thus
Now returning to the complex structure,
V can
be considered as a complex analytic manifold, and the covering
mapping V -4 V/oZ° on the quotient space
defines a natural complex analytic structure V/c°
; those coordinate neighborhoods on V
small enough to project homeomorphically to coordinate neighborhoods on
V/oZ°
.
can be taken as
V/;°
Therefore
V/;C
has the
structure of a compact complex analytic manifold; a manifold of this form will be called a complex analytic torus.
It is obvious
chat the group operations are complex analytic, in the sense that the mapping
V/ &'
x V/ Z -> V/ a'c
-137-
aerinea by V/&;°
(p,q) --1 p - q
is a complex analytic mapping.
Thus
is a complex Lie group, indeed an Abelian complex Lie group.
(Lie groups here are connected.)
As those who are familiar with Lie groups know, any compact
Abelian complex Lie group is of the form
V/o2'
.
(See for instance,
C. Chevalley, Theory of Lie Groups I, (Princeton, 1946); the discussion there is for the real case, but goes through in the same manner in the ,complex case.) V/ate[
___I V'/
'
It is obvious that an isomorphism
between two compact Abelian complex Lie groups
is equivalent to a complex linear isomorphism V --4 V'
the lattice X into the lattice
il°'
which takes
Therefore the structure we
.
are investigating is nothing more nor less than that of a compact Abelian complex Lie group.
Actually, however, the Lie group aspect
need not be considered any further; for the complex structure itself essentially carries all the information. Lemma 20.
spaces, and eX C V,
Let
V,V'
XI C V'
be
g dimensional complex vector
be lattice subgroups of rank 2g
The compact complex manifolds V/X
v'/' ° '
,
are holomorphically
equivalent if and only if there is a complex linear isomorphism
P: V
V'
such that F( ,'L° ) = ;C'
Proof.
.
First, a complex linear isomorphism
F: V -4 V'
is a complex analytic homeomorphism; and if F(X ) evident that
aiO'
it is
F induces a complex analytic homeomorphism
f: V/ T V'/ c2°j
.
Next, assume conversely that there is a
complex analytic homeomorphism
f: V/,T -> V'/X '
sition of the natural projection V --> V/a'
.
The compo-
and the mapping
f
yields an analytic local homeomorphism V --> V'/ k ' ; and since
V and
are simply connected, the latter mapping can be factored
V'
through an analytic local homeomorphism F: V --> V'
.
That is to
say, there will exist a complex analytic local homeomorphism
F: V --> V' which induces the given mapping f: V/o'Z° ---> V'/ Z' X' a
X e Z there will exist an element
so that for any element such that
F(p+ X) - F(p) + X'
(9)
In terms of coordinate systems
for all points p e V . for
V and
(w1,...,wg)
by a g-tuple
wi = Fi(p)
plex variables.
for
(z1,...) zg)
, the mapping F will be given
V'
of complex analytic functions of
g
com-
Differentiating equation (9), it follows that (aFi/azi)(P + x) = (aFi/azi)(P) X e a'
for all points p e V and all elements aFi/az3
.
The functions
are thus invariant under 7 , and so define complex
analytic functions on
V/oZ°
; but since
V/ e2'
is compact, it
follows from the maximum modulus theorem as in lemma 1 that aFi/az3
is constant.
The mapping F
being a local homeomorphism,
a homeomorphism from that
F(
e
) - 02''
V/°'_
F
is consequently linear; and
is nonsingular.
onto
V/EZ° '
,
For
F to induce
it is then necessary
, and the proof is therewith concluded.
It is then evident from this Lemma that two compact Abelian complex Lie groups
V/1L°
and V'/ oZ°1
are isomorphic pre-
cisely when their underlying complex analytic manifolds are analytically equivalent.
Therefore in future investigations the group
structure can be ignored, in part.
For emphasis, it should be
repeated that a complex analytic manifold will only be called a complex analytic torus when it is analytically equivalent to a
manifold of the form V/,Z vector space and a° C V
, where
V
is a
g
dimensional complex
is a lattice subgroup of rank 2g ; and
thus a complex analytic torus carries a unique further structure of a compact Abelian complex Lie group. It is frequently useful to be much more explicit in the So choose a
description of a lattice subgroup or complex torus.
basis for the vector space
V = Cg ; the elements of
vectors of length g .
V , or equivalently, an isomorphism .
will be written as complex column
Also choose a set of generators
for the lattice subgroup vT . represented by a column vector
Each vector Ai _ (xii),
set of all 2g of these vectors form a
X3 e a° C (l
..,
g
will be
i = 1,...,g ; and the
g X 2g matrix A = (Xi3)
called a period matrix for the lattice subgroup Z C V or complex torus V/2 °
.
Then the
2g x 2g
square matrix
where X denotes the complex conjugate of the matrix A , will be called an associated full period matrix. Lemma 21.
A complex matrix A of g rows and 2g
columns
is a period matrix for a compact complex torus if and only if its (XA)
associated full period matrix Proof.
is nonsingular.
By definition, the columns of A
of maximal rank in Cg
generate a lattice
(equivalently define a complex torus) if
-1)+0-
and only if they are linearly independent over the real numbers; therefore the contradiction of this situation is the assertion that
there exists a real column vector x of length zero, such that
Ax = 0 .
2g , not identically
If there exists such a vector x , then Conversely suppose
x = 0 , so the full period matrix is singular. (AW
that the full period matrix is singular; there will then exist a complex column vector zero, such that
z = x + iy of length 2g , not identically
(WA) z = 0 , or equivalently, such that Az = Az = 0 .
But then Ax - Ay = 0 , where
are real and not both are identi-
x,y
cally zero, and that suffices to complete the proof.
There are of course several equivalent ways of
Remark.
expressing the condition that a
g x 2g complex matrix A
be the
period matrix of a compact complex torus; the version used in the preceding lemma is perhaps the simplest to state.
The version closest
to the definition is that, for a real column vector x e Ax = 0
mpg
,
if and only if x = 0 ; this is just the assertion that the
columns of A
are linearly independent over the real numbers, and
the preceding lemma demonstrated that this assertion is equivalent
Yet
to the assertion that the full period matrix be non-singular. another version is that for a complex column vector T,
zA
is a real vector if and only if
the transpose of
z = 0 ; here
z , hence a row vector.
z e Cg , tz
denotes
To show that this assertion
is equivalent to the condition of the lemma, it suffices to show
that there exists a non-zero vector if and only if ( A ) is singular.
that iss that
tlllzA
z e Cg
such that
First suppose that
tril s for a non-zero vector /=z
-141-
tzA tzA
z ; then
is real is real,
(yz, z)
=
I
T
0,
-
lW
hence
is singular.
writing A = R + iS
where
to the condition that
(S)
Conversely, suppose that R,S
is
singular;
are real matrices, this is equivalent
is singular.
there must exist a non-zero vector
(b) a
Since this is a real matrix,
jFg
so that
(ta,tb)(") = taR + tbs - 0 ; but then Im {(tb + ita)A) = Im {(tb + its) (R + i5)) = taR + tbs = 0
which proves the desired assertion. The period matrix associated to a complex torus is not unique,
since two arbitrary choices were made; so it is important to examine the effects of these choices. the vector space
First, choosing a different basis for
V amounts to applying an isomorphism C _ e ;
representing this isomorphism by a non-singular
g x g
complex
matrix M , the period matrix is obviously transformed into the
period matrix MA , since each vector vector
M%i
.
Ai
Second, a different basis
is transformed into the Xj,...,
g
for the lattice
subgroup Z is necessarily of the form XjI - Tk Ani, , where N = (nom)
minant
is a 2g X 2g matrix of integer elements and of deter-
+1 ; this change clearly replaces the period matrix A by
the matrix AN .
Consequently, two period matrices
A,A'
represent
the same complex torus if and only if
A' = MAN where M e GL(g,C), N e GL(2g,Z) .
(10)
(Here
GL(n,R)
denotes the group of invertible n X n matrices
over the ring R .)
This equivalence relation can be used to bring
-142-
a period matrix into a simpler form, as follows.
,matrix A
into two square
Decompose a period
g X g blocks of the form A = (Al,A2)
By multiplying A on the right by a suitable integer matrix N e GL(2g,Z) , it can be arranged that the matrix
be non-
A2
g linearly
singular; to see this, merely recall that there are
independent columns in A , and a suitable matrix N can be found rearranging the columns to make these the last
the matrix A and the matrix A21A = (A21A1,I)
g
Then
columns.
represent the same
complex torus; so any complex torus can be represented by a period matrix of the form matrix.
that
g X g
identity
This is still far from associating a unique period matrix To proceed further, suppose that
to a complex torus, however.
A = (A
I denotes the
A = (A1,I) , where
I)
and
A' = MAN
represent the same complex torus, so
A' _ (Ai,I) as in (10).
N into the
Decompose
g X g
matrix
blocks
N=(AB) C DJ then (Ai,I) - M(Al,I)
Hence
A1B+D
= (M(A1A+C), M(r1B+D)) .
(
C D) is non-singular, and M = (A1B+D)-1 ; and
Al = (A1B+D)-1(AIA+C) .
The converse being apparent, it follows that
two period matrices matrices
A = (A1,I)
and
A' =(A1,I)
represent
the same complex torus if and only if there exists a matrix
N = (C D) such that
A,B+D
a GL(2g,Z)
is non-singular and Aj = (A1B+D)-'(A1A+C)
-14+3-
ivuUe UnaU, oy i.emmn et, a matirax
a complex torus if and only if
!n1,1)
is rove period matrix of
is non-singular, where
Im Al
In
denotes the imaginary part of the matrix. It is useless to proceed any further in this direction just
at the present point; but the special case
g = 1 provides an
interesting and illustrative example.
The period matrix can be
taken in the form A = (hi,l) , where
X, e C
and two matrices
A = (T,,I)
and
A' = (%I1)
and
In XI # 0
represent the same
complex torus if and only if there is a matrix
N = 1 a bd such that
'
a GL(2,Z)
aXi+ c `J = bAl + d
note that bX1 + d
3
is always non-zero, since
This is
Im Xl # 0 .
in fact precisely the equivalence relation discussed in §1(f), for the complex tori are the compact Riemann surfaces of genus cussed there.
1
dis-
Note that it is evident that there actually are
distinct complex analytic tori, so that the structure introduced is a non-trivial one.
Finally, to obtain an explicit description of the Picard
variety P(M) ti
of a compact Riemann surface M of genus
choose a basis for
g
H1(M,Z) , or equivalently, an isomorphism
as
H1(M,Z)
Z g ; the same elements form a basis for
H-(M,C) , so
there is a corresponding isomorphism H'(M,C) _ eg . choose a basis
cpl,...,cpg a r(M) 0 110)
Further,
for the space of Abelian
differentials on the Riemann surface M ; the associated cohomologyr
-144-
classes will then be of the form
rli W2 i
eWg.
kW2gi) The collection of all such vectors form a 2g x g matrix
A = (mji)
called the period matrix of the Abelian differentials on M . that by Lemma 18, if necessarily
z e Cg
is a vector such that
SEz e
g , then
z = 0 ; hence by Lemma 21 (recalling the remarks fol-
lowing that lemma), the matrix a complex torus.
is the period matrix of a compact
(Actually, to parallel the earlier discussion we
should consider the transposed matrix to rather than this merely amounts to considering the rows of 0 columns of
Note
tSl
n ; but
rather than the
, a trivial distinction which will be left to the
reader to sort out.)
The period matrix of the Abelian differentials
is of course not unique, but depends on the choices of bases for 1 (M,8)
and for
r(M, (. l'0) ; it is quite obvious that different
choices have the effect of replacing the matrix
N f M, where N e GL(2g,Z)
and M e GL(g,C) .
SZ
by a matrix
Therefore, recalling
equation (10), all of these choices lead to the same compact complex
torus, which will be called the Jacobi variety J(M)
of the Riemann
slxrface M Now the Picard variety of M
P(M)
so selecting any linear mapping
is given by
H1(M
C)
+ sr(M, 61,0) A: H1(M,C) --# Cg having precisely
-145-
as xernel,
01lM,
PP(M) - Cg/AH-(M8)
^M) is the compact complex torus
In terms of the isomorphism H1(M,C) _
.
chosen above, the mapping
g
A: Wg -> Cg is represented by a
g X 2g
complex matrix A = (xij) ; since the image of A must be all of
C9
A must have have rank g .
, the matrix
A(sr(M, OL110)) = o of
f1
is evidently that An = 0 , since the columns
span the subspace
H1(M,z) _
g C C2g
AH1(ii,z) C Cg
Br(M, C71-10) C Cog
is generated by the
where the entries of ei and since
The condition that
The lattice subgroup
.
2g
column vectors
e
,
1
are zero except for a one in the i-th place;
is just the i-th column of the matrix A , the image 2g
is the lattice generated by the
of the matrix A .
Therefore
A
column vectors
itself is a period matrix of the
Picard variety of M . In summary then, let
12
be a period matrix for the Abelian
of M is the compact
differentials on M ; the Jacobi variety J(M) complex torus defined by the period matrix
g X 2g matrix of rank g P(M)
such that
t11 .
Let A be any
All = 0 ; the Picard variety
of M is the compact complex torus defined by the period
matrix A .
(c)
Not every compact complex torus can be the Jacobi or Picard
variety of a Riemann surface; in fact, it is still an unsolved problem to describe precisely which tori arise from Riemann surfaces.
A very important partial answer is provided in the form of an additional structure which the Jacobi and Picard varieties inherit from the multiplicative, structure of the cohomology of a surface.
-146-
Recall that there is a skew-symmetric bilinear mapping
Hl(M,C) x Hl(M,C)
I?(M,C)
oh the cohomolo®r of a surface M , called the cup product.
Perhaps
the easiest way to describe this is in terms of differential forms. Under the deRham isomorphism as described in §5(a),
H1(M,C) a
(q)
r(M, £1)Idcp = 01 dr(M, c 0)
so any cohomology class in
ferential form.
If
cp,+y
is represented by a closed dif-
H1(M,C)
are two such forms, their product cp ,.'
is a closed differential form of degree
element of
Hdi(M,C)
under the deRham Isomorphism.
the cohomology class represented by cp
or '
2 , hence ]represents an
cp
'
It is clear that
is unchanged when either
is replaced by a differential form representing the same
one-dimensional cohomology class; for instance, if
then ((p+ df) ,.' = cp
* + d(f4')
.
f e r(M, 'r 0)
The mapping
(q), *) --.> cp then defines the cup product operation in cohomology.
To vary the
description slightly, consider the natural identification H2(M,C) = C as introduced in §5(a).
The cup product can then be
envisaged as a bilinear mapping H-(M,C) X H1(M,C) --- C ; and in terms of differential forms, this mapping can be described as
((p,'4')>ffm (p-* Note that the subgroup mapped into the subgroup
Hl(M,la) x Hl(M, ) C Hl(M,C) X H1(M,C) B2(M,la)
is
= R under the cup product, and
-11+7-
-H2(M,Z) = Z .
rr-.. ._.a..'
Daac DUSJ SVUt7
The cup product of cohomology classes
a,P a H1(M,C)
will be denoted by a U 0 , considered as an element of C . Choosing a basis for the group
H1(M,Z) , that is, an
'
isomorphism H1(M,Z) = S g , the cup product 2F9 X Z2g -.> Z is defined by a skew-symmetric
2g X 2g
X , called
integer matrix
the intersection matrix of the surface. M ; explicitly, if
m,n a
g
are column vectors representing one-dimensional coho-
mology classes, their cup product is the integer .tmXn .
The same
matrix of course describes the oup product in real or complex
cohomologr, in terms of the same basis. A change of basis in H1(M,Z)
is described by a matf ix N e GL(2g,Z ); and this replaces
the intersection matrix X by the intersection matrix Theorem 17.
g > 0 ; and let
t
lXH-l
.
Let M be a compact Riemann surface of genus
X be the intersection matrix and
£
be the period
matrix of the Abelian differentials on M ; in terms of some basis for
H1(M,Z) (i)
(ii)
-
Then
t11X R = 0 , (Riemann's equality); and itf X fl
is positive definite Hermitian, (Riemann's
inequality). Proof.
Let
q)1,...,(Pg a 1'(M,
.l'0)
be a basis for the
Abelian differentials on M , so that the cohomology classes Ski = (wji) a H1(M,0) = a .
form the column vectors of the matrix
Note that the differential forms
the mapping
(pi - Sq) i
cpi
are closed, and that
of (4) coincides with the deRham mapping;
the conjugate differentials
are also closed, and under the
tpi
deRham mapping correspond to the conjugate cohomology classes.
Thus
the cup products of these cohomologyr classes can be calculated by
integrating products of the Abelian differentials and their conju-
be a form of bidegree ffM cpi
cpi - cpp m 0 , since the product would
Firstly, note that
gates.
,.
(p j = 0 .
(2,0) ; therefore the cup product
In terms of the intersection matrix
X , however,
this cup product is given by
0. E (dki)k2 )2j = (toX 12)ij k,£ that is to say, is the entry in row i, column j, of the matrix t9 X 12 ; and
hence
to X 12
- 0 . Secondly, if
(p a T(M, O.l' 0)
is
any Abelian differential, then in a coordinate neighborhood Ua with a coordinate mapping ep = ha(z(P )dza , where
za that differential form can be written
ba(za)
is a holomorphic function; thus
ilha(za)I2dza .. dza = 21 ha(za)I2dxa
writing
za = xa + iya .
Since
..dye,
is the local area element,
dxa ,. dye
then for the cup product it follows that
iffMcp..cp>0, with equality occurring only when Abelian differentials, write
(p = 0 .
cp = Eicicpi
In terms of the basic for some complex constants
Cl ; so 0<
f fM c p .. c p = E cicj
i,j
I f fM (Pi
where
Plj="-ilfm -149-
cpj = E ciPi je j
i,j
fun mmt,rix
r = krij)
is clearly Hermitian, and the above inequality
shows that it is positive definite as well.
In terms of the inter-
section matrix however,
Pi3 = or in other words,
A matrix
E
k,2
(tnXS2)ij
Wki
P = itt]X 7 , which serves to conclude the proof.
Sl
with 2g rows and
g
columns is called a
Riemann matrix if there exists a skew-symmetric integer matrix X with
2g
rows and columns, such that:
tc X C = 0 ,
and
(12) itaXla
The matrix X fl
; the net
is positive definite Hermitian.
is called a principal matrix for the Riemann matrix (n,X)
consisting of a Riemann matrix
fl
and an associ-
ated principal matrix is called a Riemann matrix pair, and it is this
pair which is of primary interest.
that the set
Theorem 17 is then the assertion
(n,X) , consisting of the period matrix
n of the
Abelian differentials on a compact Riemann surface and the intersection matrix of the surface (in terms of some choice of basis for H1(M,Z)), is a Riemann matrix pair.
If
is a Riemann matrix pair, and if M e GL(g,C)
(n, X)
and N e GL(2g,Z) (Nam, tNlXN-1)
are arbitrary matrices, then clearly is also a Riemann matrix pair; this merely amounts
to verifying condition (12), a trivial calculation which will be left to the reader.
Two Riemann matrix pairs
will be called equivalent when
some matrices M e GL(g,C)
(f)X)
fl' = NfM and
and N e GL(2g,Z)
X' = .
and
(0',X')
tN l)N-1
for
Recalling equation
to
(10), it is evident that the period matrices sent the same complex torus whenever equivalent Riemann matrix pairs.
(n,X)
and
and
tn'
(n',X')
repreare
A complex torus whose period matrix
is a Riemann matrix will be called an Abelian variety; and the preceding observation shows that all the period matrices representing an Abelian variety are necessarily Riemann matrices.
Actually of
course, an equivalence class of Riemann matrix pairs determines an Abelian variety with an additional structure; such an equivalence class Will be called a polarized Abelian variety, and will be viewed as an Abelian variety with the additional structure determined by a fixed principal matrix for any period matrix defining the underlying complex torus.
With this terminology, Theorem 17 has the following
immediate consequence. Corollary.
The Jacobi variety of a compact Riemann surface
g > 0 has a canonical structure as a polarized Abelian
of genus variety.
Proof.
for
For any choice of a basis for
H1(M,Z)
r(M, 0-1'0) , Theorem 17 shows that the pair
(92,X)
and a basis
, consisting
of the period matrix of that basis for the Abelian differentials and the intersection matrix for the surface, is a Riemann matrix pair. Another choice of a basis for
H1(M,Z)
is determined by any matrix
N e GL(2g,z) ; and another choice for a basis for
determined by any matrix M e GL(g,C) .
r(M, G', 0)
is
As noted earlier, the new
period matrix is then NOM , and the new intersection matrix is tN-1)O 1
; go this leads to an equivalent Riemann matrix pair, suf-
ficing to conclude the proof.
wne aiseuss].on so far has concerned only the Jacobi variety J(M)
of the Riemann surface M , and it remains to show how this
structure is reflected in the Picard variety P(M) .
bases for H1(M,Z)
and for
matrix of the surface and
f(M, & 1'0) , let
Again choosing
X be the intersection
n be the period matrix of the Abelian
differentials on the surface; and let A be any g X 2g matrix of
rank g such that
An = 0 .
The Riemann matrix pairs
Lemma 22.
and
(n,X)
(tA, -x-l)
are canonically equivalent. Proof.
That
is a Riemann matrix pair was proved in
(fl,X)
(tA, tX i)
To show that
Theorem 17.
is an equivalent Riemann matrix
pair it is only necessary to find matrices M e GL(g,C) such that
t
A - N fl M and
and N e GL(2g,Z)
tX-l = tN-lxN l ; and selecting the matrices
M,N
in a canonical manner will show that the equivalence is canonical.
Let
P be the non-singular
be the
g X g matrix
g X g matrix P =
itn X7f , and let
Q
Q = A32 ; since An = 0 , rank A = g , and
rank (R,fE) = 2g , it is clear that the matrix
is also non-singular.
Q
The proof will be completed by verifying that the matrices
M = - i tP 1 tQ obvious t h a t
and N = X satisfy the desired conditions.
N-
1
=
tx1XXl
= t)Cl
.
For the remaining con-
dition, note that A
X
An
-/
(An
A75
1\
0r QfI
`A
0
and therefore
1 A/ A
-152-
It is
s
'
vViLOCY=11 U.Ly
tMtatN
tMtnx
Ti ;
t tfx(a,j)`
4-l so
Tl A Ql A
=tMiPQln=n,
A
NnM = tA , as desired.
This lemma shows that the Picard variety P(M)
Jacobi variety ,(M)
and the
of a compact Riemann surface are canonically
isomorphiq as compact complex tori, hence as compact complex Lie groups; and thus that
variety also has a canonical struc-
ture as a polarized Abelian variety.
It is convenient to retain
both of these names, with their oonnotations of different simple
standard representations of the same Abelian variety P(M) = J(M) ; M,
IV
and it is important for later purposes to note the explicit form of the isomorphism between these two representations. J(M) (z)
P(M)
N
as the quotient of the vector space
Cg of column vectors
by the lattice generated by the columns of as the quotient of the vector space
Considering
to
, and considering
Cg of column vectors
(w )
by the lattice generated by the columns of A , write
.T (M) °
eg:
(z)
,
P(M) =
gw)
The isomorphism is then given by
(13)
w= - iQPlz.
The verification is straightforward, and will be left to the reader.
that, for a suitable choice of basis for
H1(M)Z) , the intersection
matrix of the surface has the form
to
-I
I
0
X = J =
(14)
where
I denotes the
g x g
,
identity matrix and
the
0
g x g
zero matrix; a basis of this form will be called a canonical basis for the surface.
(The notation J or Jg will be reserved for
matrices of the form (14).)
A Riemann matrix pair defining the
(polarized) Jacobi variety of a compact Riemann surface can thus
always be taken in the canonical form n
g x g matrix blocks of the form
into
. /Splitting the matrix
(n,J)
, the Riemann
fl = (
/
conditions take the form (i)
tn2fl - tnla2 = 0 , and
(15) (ii)
t
i( SE S2
2 1
-
t
A T2) 1 2
is positive definite Hermitian;
this is \an easy calculation, which will be left to the reader. n =( S2 1
is a Riemann matrix with associated principal matrix J
if and only if (15) holds.
Note that the matrices
both necessarily non-singular.
For if
exist a non-zero constant vector
then
tc
Thus
t0'2 = .52a = 0 , hence
n2
c e 09
itc(t9251
and
11;
are
were singular, there would such that t
-
a1
"192)
n2c = 0 ; but
= 0 , contradicting
the positive definiteness of the matrix (15ii). As in the earlier discussion of the period matrices of general compact complex tori, so in the discussion of Riemaui matrix pairs, it is of some interest to examine simpler representatives of of an equivalence class.
If
J
is the associated principal matrix
of one representative of the equivalence class, then we can restrict consideration to those Riemann matrix pairs of the form
in
(11,J)
the given equivalence class; and since the principal matrix is fixed, it suffices merely to consider the Riemann matrix most general Riemann matrix class is then of the form
11'
in the some (restricted) equivalence
(or equivalently,
Be GL(2g,Z)
The set of all matrices
such that
group called the symplectic group of rank
J = tNJN)
tNJN = J
form a
g , and denoted by
Ffriting fl = (Gnl 1l , the matrix
SP(g,Z)
The
alone.
n' = NfM , where N e GL(2g,Z) ,
tN 1JN l = J
M e GL(g,C) , and
f
A2
is non-singular, as
2 11122-1
noted above; hence
11' =
is an equivalent Riemann matrix.
Thus, an equivalence class of Riemann matrix pairs with principal
matrix J
always contains a Riemann matrix of the form
12
=(I
I
.
In this case, it is readily seen that equations (15) take the form
1 (i)
is symmetric, and
111
(16) (ii)
is positive definite.
In n3'\
Thus
J
12 = l I1 f
is a Riemann matrix with associated principal matrix
if and only if
and
12'
exist matrices
121
Now suppose that
n =
nl I
are two equivalent such matrices; there must then N e SP(g,Z)
And writing N = (C
(16')
satisfies (16).
and M e GL(g,Z)
such that
at - NOM .
, it follows as in equation (11) that
Ll
111 = (An1 + B) (Cn1 +
thus two Riemann matrices
( 1), C 21/
D)-1
with associated principal
matrix J areequivalent if and only if there exists a sympeetic
-155-
C D)
Holds.
es ucu coati k.Lo
Note that, as a con-
sequence of the obvious observation in the preceding paragraph,
Cal + D is always non-singular.
The set of matrices
a1
satisfying
(15) is called the Siegel generalized upper half space, and denoted
by j g
.
The symplectic group
formations on
g
(
SP(g,z)
acts as a group of trans-
, and the quotient space
('g/SP(g,z)
is in one-
to-one correspondence with the equivalence classes of Riemann matrix pairs
(n,J)
.
This leads to a fascinating area of study, but there
is not enough time to continue further in this direction here.
Let
it suffice merely to remark that the symplectic group acts discontinuously, and that the quotient space is a complex analytic space
having the same dimension as
(q g ,
namely 2(g + 1)
.
Note that it
is clear from this discussion that not all complex tori are Abelian varieties.
(d)
It naturally occurs to one to enquire about the relation-
ships between the complex structure of a Riemann surface M
and
the complex structure of the Picard or Jacobi variety of M .
To
pose a precise question, for any two points associated point bundles _
ql
cp,
q
p,q e M consider the
; the complex line bundle
has Chern class zero, hence can be considered as an
element of the Picard variety P(M)
M X M --4. P(M) , defined by
.
There is thus a mapping
(p,q) - psg
e P(M)
; and the
question is whether this is a complex analytic mapping. Lemma 23.
g > 0
On a compact Riemann surface M of genus
consider any two points p,q , and the complex line bundle
-156-
pt-l a P(M) ; and let from, the point
q
r
be an arbitrary differentiable are
to the point p
in M .
Then representing the
Picard variety in the form H1(M,C) Hl(M,z) + 5r(M,
P(M) -
the line bundle
g e P(M)
can be represented by a cohomology class
such that
c E Hi(M,C)
au (&p)=ITw
(17)
for every Abelian differential W e r(M, Proof. are
T
(91,0)
To begin, suppose that the points
z0
can be talon to be a homeomorphism of an
open neighborhood of the point net closure
line C , mapping UO such that
to a closed disc.
T C VO C V0 C UO .
with the properties that:
the covering 14
(i)
U0 C M into the complex V0 C M
Choose an open set
Further, choose an open covering
of M consisting of the set UO
C .
p,q , and the
lie in a contractible coordinate neighborhood UO C M ; the
coordinate mapping
(ii)
'0)
and some additional sets Ua fl V0 = 0
for a # o ;
is a Leray covering for the sheaves
(Ua)
and 6- and
(Recalling §3(e) and §3(f), it is only, necessary to require
that the sets Ua be coordinate neighborhoods, and that these sets and all their intersections be contractible.)
c(g) = 0
Since
C*)
the line bundle
g
can be represented by a cocycle
formed of constant functions. a meromorphic section ,,, (fa) =
in Ua fl 0 .
And since
(fa) a r( M *(g))
the functions
(ga$) E Z1(,
g = ptg1 , t#tere will be with divisor
fa thus satisfy
fa = gad f
In each set Ua for a # 0 the function
fa
is
nolomorphic and nowhere vanishing; selecting some branch of the
logarithm, introduce the holomorphic functions
as=-2tt1 logfaeMa,a) In the annular region U0 - VO
.
the function log f0 will also be
single-valued, since the sum of the residues of the function
d log f0 within V0 the set
is zero; so select a
U0 which coincides with
region UO - VO .
CO*
(For instance, select a branch of
in an open neighborhood of the point set closure
multiply it by a
C`O
- 2ai log f0
U0 - VO , and
function which is identically one on UO
aa6 = a0 - as
in Ua fl u, , it follows that
- VO
Then
and which vanishes identically on a suitable subset of VO .)
defining
in
in the annular
log fO
-
o0
flmction
exp 27ri aC _
faffP = ga$ ; these functions are therefore constants, so form a cocycle
(aa$) e Z1(Vt C)
mines a cohomology class line bundle
Recalling §8(a), this cocycle detera c g1(M,C)
g
a u (5p)
To calculate the cup product class with the cohomology class
5p e H1(M,C)
(daa) a r(M, Ee1)
5q'
tp
.
Since both
cp
represents the cohomology
and daa are holomorphic
differential forms, hence differential forms of degree
everywhere outside of the set UO , it follows that M - U0 ; so that
a'u 5cp = ff,0 da0
-158-
a under
Consequently
under the deRham isomorphism.
a u 5q a 11M daa A cp
represented by an
represents the cohomology class
the deRham isomorphism; and obviously, class
of this cohomology
cp a r(M, L-l'0) , observe that the differential
Abelian differential form
which represents the complex
cp
.
(1,Q)
daa ,.
tp = 0 on
Select a holomorphic
function h
in the set U0
so that
dh = c ; then from Stokes'
theorem,
a u 5P = ffU da0 ,A dh = - f f
0 =
The function
d(hda0)
U0
ff hda0
a0 was chosen so that it coincided with
- 2,1-Ri log f0
along the boundary 3U0 ; so applying the residue theorem, and remembering the definition of the function h ,
au 5P_-f hdv0=12- f hdlogf0 0
0
= h(p) - h(q) = f dh = f c T
T
This demonstrates the theorem in the special case that lie within a coordinate neighborhood. be split into segments qi' qi+1'
and
Ti
Ti
In general, any arc
from points
qi
can
qi+l , such that
representing the line
ai e H1(M,C)
bundle
ai u 5p = fT p .
Cl
to
T
lie in a coordinate neighborhood; there will
then be a cohomology class t
p,q, and T
, so that
qi+l qi
The sum
i
a = Eiai a Hl(M,C)
tql = tptgl
represents the line bundle IIitq i+1
i
and
a U 5cp = E1(ai u 5q) = Ei fT T = fT u , which completes the proof in general. Now choose a basis for Hl(M,C) _ C
,
H1(M,Z) , thus identifyizig
H1(M,Z) = ig ; and choose a basis
cpl, ...,cpg a P(M, 6L-110)
for the Abelian differentials on
M.
X be the intersection matrix of the surface M in this basis,
Let
fl
be the period matrix of the Abelian differentials on M , and A
be any g x 2g matrix of rank the non-singular
g
such that An = 0 ; and introduce
g X g matrices
P- itaXSF,
Q=All.
The Picard-Jacobi variety of the surface M then has the standard representations
J(M)tg,
ti(M)&g
which are related as in equation (13).
With this notation, and the
convention that all vectors are viewed as column vectors, the following holds. Theorem 18.
are from a point
q
(Abel's Theorem)
to a point p
M , and introduce the vector
Let
be any differentiable
T
on the compact Riemann surface
t e CF with coordinates
i=1,...,g .
I
ti
pC-1
Then the complex line bundle
e H1(M,
*) , considered
as an element of the Picard-Jacobi variety .1(M) = P(M)
of the
surface M , is given explicitly as
t e Cgftfg = J(M) , - i QF-1t Proof.
Let
e
C9/AZ 2g
= ti P (M)
.
v = (si) a H1(M,C) = C 2g be a cohomology
class as given in Lemma 23; thus
ti = f i - a U (eq'1) _ T
or in matrix form
-16o-
E
k'A
sk
(19)
Recalling that
tX = -X , and that
(S2,3f)"1 = I
Q"
1.
n
f
as in
A
Lemma 22, rewrite (19) in the form
t = - tnX(fl,n)
Now by construction, line bundle
s = (0,iP) \ Q"1. A / As a Cg/Ag = P(M)
Q_1 A) s =
(
iPQ-1As
A
represents the complex
g ; that is to say, I = As = - iQP-lt a CF/AZ2
= ti(M)
which is the second assertion of equation (18).
,
The first assertion
of equation (18) then follows immediately from equation (13), and the proof is thereby concluded.
There are a number of almost immediate consequences of Abel's theorem, as stated here. Corollary 1.
M --> P(M)
defined by p --
Proof.
M -> 3(M)
q e M , the mapping
For any fixed point
pCl
is a complex analytic mapping.
It is more convenient to consider the mapping
determined by the canonical isomorphism r(M) = P(M)
By Theorem 18 this map has the form
p E M --4
(fp (p
)
e
fg/t 2g = J(M)
The value of the integral is obviously a complex analytic function
of the upper end-point p , and hence the corollary follbws at once. Corollary 2.
For any points
choose differentiable arcs
Ti
from
-161.-
pip ...'pr'g1,...,gr E M , qi
to
pi
on the surface M .
Then
Pi
_ t
... t
Pr
if and only if the vector
...
ql
t e
qr
with coordinates
r
ti to 29
belongs to the lattice Proof.
J
1
1 Ti
i
.
By Theorem 18 the line bundle
tql
gJ = 3
as an
J
element of the Jacobi variety in given explicitly by the vector having coordinates
tJ e r-g/tIM2g - J(M)
tij = fT
hence the bundle
is given explicitly by the vector
... it
I
Ti ;
r t
=1
NOV
t iJ
if and only if
...
P1
ql
pr
in Cg/t1Mg , that
which corresponds to the condition that t = 0 is, that
t e
tg
, as desired.
Fbr any points
Corollary 3.
choose differentiable arcs
Then
... t
.,qr E M qJ
to
pi
on the surface M
if and only if there is a close.
5
q1 T
from
TJ .
1 Pr differentiable loop
g -1
qr
qr
on the surface M such that
r z fTq) =ITfQ j
J=1
for all Abelian differentials Proof.
Let,...,
cp a P(M, (511,0)
g
be closed differentiable loops
on the surface M which generate the homology group H1(M,Z) and choose the dual basis for the cohomology group H1(M, . for the Abelian differentials
(pi a r(,
Then
it follows that
i
By Corollary 2,
for
q)
Cdki = fIrk
1=1,.. .,g
= tql ... 5qr
61 ...
and
k:mL,...,2g
.
if and only if there is
SPr
an integral vector n - (nk) e Z 2g such that
2g
r
1 fT, cpi == kl nkWki =
4 Ti
J
2g where
T = Z nk
Since
.
form a basis for the Abelian
cpi
k=1 differentials, the latter condition is equivalent to the condition that r r
z
J=1 for all Abelian differentials Recallin, that
fT
fT
cp , as desired.
... 5
1 there is a meromorphic function
9' (f)
(p
j
C1
..
pr ql
t-1 = 1 qr
if and only if
f on the surface M such that
r r E 1'Pi - E 1.q1 , i=1
i=1
Corollaries 2 and 3 can be restated as necessary and sufficient conditions for the existence of such meromorphic functions; this is the traditional form in which Abel's Theorem is stated.
-163-
12, (a)
There are interesting structures which are finer than com-
plex analytic structures on manifolds, but which play an important role in complex analysis.
Recall from the discussion in §1(b) that
the important property of complex analytic functions, for defining a complex analytic structure, is the pseudogroup property: the composition of two complex analytic local homeomorphisms is again a complex analytic local homeomorphism whenever the composition is If a subset of the set of complex analytic local homeo-
defined.
morphisms has the pseudogroup property, then there is a further structure on complex manifolds associated to that subset.
Perhaps
the most interesting such subsets are defined by differential equations; these are the only such subsets which will be considered here.
Suppose that
f: U ---> V
is a complex analytic local
homeomorphism between two open subdomains U,V of the complex line
C ; the condition that
course just that
f' (z)
0
the differential operators
f be a local homeomorphism is of for all points
z e U .
Introduce
e1,62 , defined as follows:
eif(Z) = fz
(1)
e2f(z) =
(2)
2f'(z)f"(z) - 3f"
(.)2
2f'(z)2
Since
f'
is nowhere vanishing in U , the functions ef are
holomorphic throughout U .
The differential operators
0V
are
of particular importance for their behaviour under the composition of mappings.
-164-
Lemma 24.
f: U - V and g: V - W be complex
Let
analytic local homeomorphisms between subdpmains of the complex
line C , and let
h - g a f be the composition of the two.
Writing w - f(z) , evh(z) =
(3)
Proof.
Since
evf(z)
for V=1,2
and ds = f'(z) , it follows
h(z) = g(w)
from the chain rule for differentiation that h'(z) = g'(w)f'(z)
,
h"(z) - g"(w)f'(z)2 + g'(w)fu(z) gm(w)f,(z)3 + 3g"(w)f'(z)f"(z) + g'(w)fm(z)
h'"(Z) =
.
Thus
g.,;w)fg(z)2 t
w)f"(Z)
elh(z) =
= elg(w).f'(z) + e1f(z)
and
(z)-2g'f'(97W)3+3g"f'fm+g'f".1-3[(97)2(f') +2g'g"(f.')2ff+(g')2(f,)2
02h
2(g' )2(f,)2 =
92f(z)
,
which complete the proof. Now let homeomorphisms is defined.
J v
;'V
be the family of all complex analytic local
f such that
0Vf(z) = 0
at all points
z
where f
It is an obvious consequence of Lemma 24 that the family
has the pseudogroup property; this introduces the new structures
next to be investigated.
(It naturally occurs to one to ask why
these two differential operators are selected; the reason is that they are essentially the only such operators.
-165-
More precisely,
one dimension, defined as the set of solutions of a system of differential equations involving only the first and higher derivatives, and possessing the pseudogroup property; then either
a _ 9 1 or 4.92 .
This is not really difficult to show,
but is too much of a digression to enter into here; it is of course part of the general problem of classifying pseudogroups.
For
further discussion, see for instance Elie Cartan, Sur la structure des groupes infinis de transformations, Ann. Be, Normale, 21(1904),
153-206; or the paper of Guillemin and Sternberg referred to earlier (page 4).)
The families
that
f(z) = az + b
fl(z)
are in fact very familiar, in a more
Firstly, if
explicit form.
necessarily
9'v
0
f e R l , then
f"(z) s 0 , so that
for some constants
is merely that
the complex affine mappings.
a
a,b ; the condition
Thus, a 1
0 .
consists of
Secondly, it is an easy calculation
to show that
f'
82(z) _ - 2 fI (z)j
(It)
dz
(z)-*
-2
Therefore, if
f e 9
fl(z) _ (cz + d) -2
2
for some constants
f'(z)_
77 dz
it follows that
s o , so that
c,d ; and integrating again,
necessarily f(z)
for some constants
a,b,c,d .
merely to the condition that
az +b
ez+d The condition that
ad - be
the complex projective transformations.
0 .
f'(z)
Thus, 9 2
0
amounts
consists of
(These are also c4lled the
11near iract].onaJ_ or Mooius transrormations oy some writers. ) me differential operator
is also called the Schwarzian derivative.
B2
Turning next to the associated structures on manifolds, let
M be an arbitrary two-dimensional topological manifold, and let
be a coordinate covering of M , with the coordinate
(Ua,za)
transition functions
za =
The coordinate covering
covering
in the intersections
(Ua,za)
will be called an -9* v
Ua fl UP .
coordinate
if all the coordinate transition functions belong to the
family 3 v .
Two a v
coordinate coverings will be called equi-
valent if their union is also an 3
coordinate covering.
v
a consequence of the pseudogroup property of 3 v
It is
that this is
actually an equivalence relation; recall the discussion of complex analytic structures on page 3.
An equivalence class of 9L v
dinate coverings will be called an a v
coor-
structure on the manifold.
The adjectives affine and projective will frequently be used in
place of
3-
1
and a2 respectively, in view of the explicit
form of these families of mappings; thus an affine structure is
an 3 1 structure, and a projective structure is an a2 structure.
Note that an affine coordinate covering is also a projective
coordinate covering, since
C 32
; and that two equivalent
affine coordinate coverings are also equivalent when considered as projective coordinate coverings, for the same reason.
Hence an
affine structure belongs to a well defined projective structure; the affine structure is said to be subordinate to the projective structure.
In the same manner, a projective structure is subor-
dinate to a well defined complex analytic structure.
-167-
For this
__... ..a ..aaaUG ous LN:4LLLeh on a Riemann surface, meaning projective or affine structures sub-
ordinate to the given complex structure.
Observe that it remains
to be seen whether a complex structure actually has a subordinate
projective or affine structure, and whether that subordinate structure is unique. To investigate these questions, consider any complex analytic [Ua,za)
coordinate covering
from
of the Riemann surface M .
§5(c) that the canonical bundle
K e RZ(M, &*)
Recall
is defined
there %p(p)
by the cocycle (Kc¢) a Z1( 1/t MB(z0(p)-1
p e Ua n U. , and fQ*
for points
transition functions.
To each intersection Ua n UP
complex analytic function and consider the element
efC43 (z,)
zP
naturally
Thus in terms of the local coordinate
(zP(P)) = eyfC'0(e (p)) ;
a
and if p c U,, zy,
-(Kv))
defined in U0 ,
(5)
mapping
associate the
defined in yUU n UP) ;
ava, a T(Ua n UP,
associated to that function.
mapping
are the coordinate
as well, then in terms of the local coordinate
defined in UY , av
(zy(p)) =
The coordinate transition functions satisfy the condition that
f ey (zy)
= f a d o fly (zy ) for zy, e zy,(Ua n U0 n Uy,) ; so by
Lemma 24,
eVf
(zY)
= eVfV(zd f1ay(zr)v + eVffy,(z,,)
.
Rewriting the latter equation (za(p))-K07(p)-V
Q,,a,(zy(p)) = aV
(6)
+ aVa7(z7(p)) ;
or equivalently,
(z7(p)) + oVPY(z7(p))
aVCP (z7(p)) = a 4r
However, this means that the elements (aVa0) e'Z1(Vt
,
6QKV) )
.
(ate)
define a cocycle
Thus to any complex analytic coordinate
covering b7 of M there is canonically associated a cocycle
(avai) =(e fo'd e Z1(LM, e (KV)) An 4
V
V=1,2.
connection for the covering Ut is a zero cochain
_h = (ha) e co(V1 h,h'
,
(KV))
such that
for two coverings M , Z41
8h = aV .
The connections
will be called equivalent if to-
gether they form part of a connection for the union of the two coverings.
(Note that the cocycle
aV
associated to the union of
the two coverings consists of the original cocycles for the two separate coverings, in view of the fact that it is canonically defined. h,h'
The equivalence condition is just that the two connections
can be extended to form a connection for the union of the
two coverings.) An equivalence class of connections will be
called an 3 V
9
1
13 2
connection for the manifold M .
As before, an
connection will also be called an affine connection, and an connection will also be called a projective connection.
Explicitly, an a V (ha) a r(Ua, 4 (KV))
connection for VI
consists of sections
such that
ova (P) = hf3(p) - ha(P) for p e Ua fl u,3
-169-
.
-
za in the coordinate- neigh-
- .. . .. .6%J16aW. UUAPFling xuncti1on borhood Ua , the section ha function ha(za)
in
is realized as a complex analytic
za(Ua) ; and the coboundary condition can be
restated as
(7)
avC43 (z0(P)) - h0(z0(P)) - KI(P)-Vha(za(P))
p E Ua n UJ .
There is a canonical one-to-one correspondence
Theorem 19.
between the a
for
connections on a Riemann surface and the
V
aV
structures on that surface. Proof. M ,
Let
h be an
av connection on a Riemann surface
and choose a representative connection
(ha) E C°(Vt , B (KV))
of M.
for some complex analytic coordinate covering V = (U(,za)
Note first that, after passing to a refinement of the covering if
necessary, there will be complex analytic homeomorphisms wa on the sets
za(Ua) = VaC C such that
ha(zes) = evwa(za)
To see
.
this, it is only necessary to show that in some open neighborhood
of any point there will exist a solution wa of the differential equation
eVwa = ha , such that wa # 0 .
For the case
v = 1
the differential equation is the linear equation w'a - haws = 0 which has solutions with arbitrarily prescribed values for wt - at any point.
For the case
V = 2 , recalling formula (4), the dif-
ferential equation can be rewritten da + 2hava = 0 where w'a = va , and the same result holds.
Note further that the most
general such homeomorphism is of the form wa = va , wa for some
element vas & satisfying
.
v
For if wa is any analytic homeomdrphism
ha(zes) = evwa(za) , then putting va = wa V. wal
write wa = va a wa ; but by Lemma 24,
ha = evwa = ev(va 0 wa) _ hence
evwot =
hot
and va E
evvv = 0
Now for the given open covering
(Ua) , the most general
complex analytic coordinate covering is of the form f o r some c o m p l e x analytic homeomorphisms
(U
za)
wo, a
the
trot: V a --> W o e C ( V
associated coordinate transition functions are
N
foo = (we a za)e NO a z0)
= wa a fcS a w;l , where fcS are the
coordinate transition functions for the covering
(U.,za)
.
Writing
and applying Lemma 24 again,
Wa a fco = fC4,3 a wo
(evwa)(faO)V + (evfoo)
(ejC4)(w;)v + (evw,)
=
or upon rewriting, (Bv? a3)(w0)V =
avQ
+ ha
h0
N where
hot = evwot .
From equation (7) it then follows that
precisely when ha is a connection for the covering VI each 14
v
.
evfa = 0 Thus
coordinate covering corresponds to an av connection;
and from the observations in the preceding paragraph, this is a one-to-one correspondence.
It is obvious that equivalences are
preserved, hence the theorem follows as stated.
An 3 v
Corollary 1.
coordinate covering Ut = (Ua,za)
on a Riemann surface M represents the a v essociated to the
V
connection h
sented by the zero 3v connection covering v4
if and only if h (ha) e Co(U
,
is repre-
(KV))
for the
.
Proof.
connection h
structure canonically
Since the structure canonically associated to the is described in terms of an analytic coordinate
covering
by a change of coordinates by homeomorphisms
(Uce za)
satisfying
wa
evwa = ha , this is entirely obvious.
Note that if h = (ha)
is an
connection, then from
9°v
equation (7) it follows that the most general av connection is
h+ g where
set of -
g e r(M, 8 (KV))
is an arbitrary section; thus the .
connections, if non-empty, form a complex linear maniy(KV)
fold of dimension
.
And applying Theorem 19, if a Riemann
surface M admits any 3v structure, then the set of all structures form in a canonical manner a complex linear space of dimension
y(KV)
For affine structures this dimension is
.
y(K) = g ; and for projective structures this dimension is 7(K2)
3g - 3 , by the Riemann-ROch theorem.
As for the existence of an 3'v
connection, it is clear
from the definition of a connection that the necessary and suffi-
cient condition is that
av = 0
in
Rr'(M, (9 (KV)) , where
ava¢ = evf
the cohomology class defined by the cocycle
ay
is
a z1(Ul, ({ KV)).
Recalling the preceding investigations of these cohomology groups, the following existence theorems arise. Corollary 2.
A compact Riemann surface of genus
g > 1
always admits projective structures. Proof. R-(M,
By the Serre duality theorem
-(K2)) = r(M, D (K-1)) ; but since
it follows that c(K) < 0
for
c(K-1) = -c(K) = 2 - 2g
g > 1 , and therefore that
R1(M, 6-(K2)) - r(M, &(K-1)) - 0 , which suffices to prove the assertion.
-172-
Corollary 3.
A compact Riemann surface admits affine
structures if and only if
c(K) - 0 , hence if and only if the
surface has genus one. Proof.
By the Serre duality theorem again,
is canonically dual to
H1(M, m (K)) = C
r(M,
'.ice
; hence there is an isomorphism
Q."
Select a coordinate covering
.
that the cohomology class
cocycle
H1(M, &(K))
a1 E H (M, 6(K))
_ (U(X,za)
so
is represented by a
E Z1(Vt, 15-(K)) ; recall that
alb = elf C43
3 f - dz0 log f'
0-1,0)
Considering aB$ E Z1(R ,
,
dz
log
then a100 dzP = -d(log Kim)
$ut this is the same form of cocycle considered in the proof of Lemma 14+;
so, applying the arguments there (especially on page 102),
it follows readily that under the chosen explicit form of Serre's duality, the cohomology class - 2iri c(K) .
Hence
a1 = 0
a1
corresponds to the constant
if and only if
c(K) = 0 , which
suffices to prove the assertion.
It was demonstrated earlier (page 115) that p1
is the
only compact Riemann surface of genus zero, and it obviously has a projective structure; this case is essentially trivial, and will henceforth be excluded from consideration.
The Riemann surfaces
of genus 1 are the only compact Riemann surfaces which admit affine structures, as a consequence of Corollary 3. seen directly as follows.
If
(Ua,z()
This can actually be
is an affine coordinate
covering, the transition functions are of the form
-173-
za = azp + b
for some constants
a00,b00 e C ; and so the canonical bundle is
defined by the functions
KC49 = (dza/dzo)-1 - a
.
As in §S(a),
line bundles defined by a constant cocycle necessarily have Chern class zero, hence
and so
0 = c(K) - 2g - 2
g = 1 .
An affine
structure is itself a projective structure; and we shall later see that in this case there is a one-to-one correspondence between In general, for Riemann surfaces
affine and projective structures. of genus
g > 1 , there are no affine structures at all, by
Corollary 3; but each surface admits a family of projective structures, by Corollary 2.
(b)
The families ,3-v
merely as pseudogroups.
can be considered as groups, rather than In view of Corollary 3 of Theorem 19, for
the remainder of the discussion here we shall consider explicitly
only the family
of projective transformations; the reader can
readily provide corresponding statements for the family affine transformations.
of
D-1
Viewing a projective transformation as a
complex analytic homeomorphism
cps
]P ---> ]P
(as discussed in
L. Ahlfors, Complex Analysis, (McGraw-Hill, 1966), for example), compositions are well-defined for any two projective transformations, so the set of all such form a group; this group is called the projective linear group of rank 2 over the complex numbers,
and will be denoted by PL(2,C)
.
The projective structures on a surface can in a sense be described by a slight modification of the cohomological machinery
which has been used earlier.
Let V& = (TJ.)
of the topological surface M , and
be an open covering
G be any abstract group,
(not necessarily commutative).
coefficients in
G
is a function
0
q
= P(Ua ,...)u
q o be denoted by Cq(1 o
e G ; the set of all such q-cochains will
)
9
G) , but this is now viewed merely as a set,
A one-cochain
with no specified group structure.
IPa = p,, and
is called a one-cocycle if Ua n U. n U7
(cpaa) a Cl(UZ,G)
%6107 = 'Pay
whenever
0 i and the set of one-cocycles will be denoted by
Z'(V(.,a) . Two cocycles
will be called equivalent
((pCO))(*M)
if there is a zero cochain
(ea) a C°( It ,G)
such that
e0- l ; the set of equivalence classes will be denoted by
*CO = eacp
H1( 1,G)
which associates to any
qp
(Ua ,..., a) a N(an element
q-simplex Ta ...a
A q-cochain of the covering It with
,
and will be called the one-dimensional cohomology set
of M with coefficients in
G .
If 2P is a refinement of the
covering X , with refining mapping
g:
) -> l
, then as in §3(b)
there are induced mappings µ: Cq(y1 , G) -> Cq(V , G) .
It is
easy to doe that these lead to a mapping µ*: HI'(VL , G) -> H'(( , G) verification will be left to the reader.
of Lemma 5, if w
-'? I,
Further, as an analogue
V: ' -> l are two refining
and
(For if ((pp) _ ((p(U(,,,u )) a Zl(Vt,G)
mappings, then µ* = V* .
then define a zero-cochain (a) e C°(Ir ;G)
6a = e(a)
p(µ ,,vva) .
Then
µ**p(Va,VV) =
tp(µVa,VVa)cp(VVa,vVP)T(v VO,µVo) µ*(p
by
a .v*v(Va,V0).e
,
so that
is equivalent to v*(p .) Then put H'(M, G)
= dir.lim.
Y.
Hl(29 , G)
to define the first cohomology set of M with coefficients in the group
G .
-179-
,
Lemma 25.
There is a canonical mapping from the set of
projective structures on a surface M into the cohomology set Hl(M,PL(2,C))
.
Proof.
For air projective structure select a representative (Ua,za} , with coordinate transition
projective coordinate covering functions
((p
M
)
.
The elements
associated to non-empty intersections
(P. = PL(2,C)
In either case
Ua n UP C M . ever Ua n U
.
If
(Ua,za)
, and
`Poo _ lp
(poPq)07 = `pay
when-
((p) determine a cocycle in
n U7 # 0 , so that
zI(Vt ,PL(2..C))
can be considered in two
qqa$: z0(Ua n u,) - za(Ua n U0) , or as
ways: either as mappings elements
(per
is an equivalent projective coor-
dinate covering defined in terms of the same open covering Vt of M and having coordinate transition functions are elements cycles
((pad)
sa a PL(2,C)
and
(pa13 )
such that p
ati
are equivalent.
((PCO) , then there
= B q
a
0-1
P
;
so the co-
There is thus a well-
defined mapping, from equivalence classes of projective coordinate
coverings of M defined in terms of the open covering Vi., into the cohomology set H1(Zn ,PL(2,C)) covering
(U.,za)
.
A projective coordinate
induces a natural projective coordinate covering
for any refinement If < 1)L , and this is evidently compatible with
the cohomology mappings H1( Dt ,PL(2, C)) --- H1(]I' ,PL(2, C)) . Finally, two projective coordinate coverings (Ua,za)
and
(UU,za)
are equivalent if and only if they induce equivalent projective
coordinate coverings for a common refinement V of Vt and This serves to conclude the proof.
The element of
H1(M,PL(2,C))
corresponding to a projective
structure will be called the coordinate (cohomology) class of that
structure.
The mapping which associates to a projective structure
its coordinate cohomology class to-one nor onto.
a
H1(M,PL(2,C))
is neither one-
However, restricting consideration to the pro-
jective structures subordinate to a fixed complex structure, the mapping is one-to-one, in the sense that two projective structures on a Riemann surface are equivalent when they have the same coordinate cohomology class.
Before turning to the proof of this
assertion, it is convenient to introduce some further terminology. Again consider an abstract group
G , but now suppose that
G acts as a group of homeomorphisms on a topological space For any cohomology class
cp E Rl(M,G) , select a basis
S
.
V& = {Ua)
for the open sets of the topological surface M and a representative
cocycle
((pao) E Z'(VL G)
for the cohomology class
To each set Ucc a , associate the set mappings from Ua into associate the function
a of continuous
S ; and to each inclusion Ua C U0 IJP which takes a mapping
pPa: ,da
za E ,0 a into the mapping
E
,d
0 defined by
(Poaza)(P) = WOa(za(P))
Since
cp
for
p e Ua C U
is continuous, this definition makes sense.
cpPa: S ---- S
Whenever Ua C U0 C Uy
and
za a e a , it follows readily from
the cocycle condition on (T.)
that pyppp = pya .
Therefore
( Vi , AM,pa13 ) is a presheaf of sets over M , which is easily seen
to be complete; the associated sheaf will be called the sheaf of germs of continuous sections of
denoted by 9c ((p,S)
.
cp
with values in
S , and will be
A section z = (za).E r(M, 4 (cps)) corre-
sponds to a family of continuous mappings
2 a: Ua - S
such that
za(p) _ % (zo(p)) structure, and
whenever p e Ua n UP .
If
has a complex
8
G acts as a group of complex analytic homeomor-
phisms Of S , then in the same manner we can define the sheaf of
germs of complex analytic sections of T with values in S , a sheaf which will be denoted by structure, and
d-((p,S)
If S has an algebraic
.
G acts as a group of automorphisms of that struc-
ture, then the sheaves
9((p, S)
and e-((p, S)
can be given the
structures of sheaves of those algebraic structures.
The space
S
will be dropped from the notation if there is no danger of confusion. Two examples will be of particular interest here. in which
and
G = GL(n,C)
S = Cp ;
G
acts as a group of complex
analytic isomorphisms of the complex vector space mology class
cp a 111(M,GL(n,C))
The first is that
6t
.
A coho-
will be called a flat complex
vector bundle of rank n over M ; the corresponding sheaf t(gp,C$) of complex analytic sections has the structure of a sheaf of com-
The case n = 1
plex vector spaces.
is just the case of flat
complex line bundles, as considered earlier. SL(n,C)
which
can be used in place of GL(n,C) G = PL(2,C)
and
S = P .
.
Of course, the group The second is -that in
A cohomology class
(p a H1'(M,PL(2,C))
will be called a flat projective line bundle over M Note, by the way, that if
(p a H1(M,PL(2,C))
is the coor-
dinate cohomology class of a projective structure on M , and if VC = (ii,za)
is a projective coordinate covering with coordinate
transition functions eohomology class the sections
(cp a$) a Z1(UZ ,PL(2,C))
cp , then actually
which represent the
(za) e r(M, a ((p,P ))
.
Indeed,
za have the further property that they are local
homeomorphisms from M into P .
Conversely if
(p a H1(M,PL(2,C))
has sections
(za) a r(M, C ((p,]P))
which are local homeomorphisms,
then those sections define a projective coordinate covering of M upon suitable refinement, and
cp
of that projective structure.
The subset of
is the coordinate cohomology class r(M, e (c),l?))
con-
sisting of sections which are local homeomorphisms will be called the set of coordinate sections, and will be denoted by ra(M, a ((P,1P)).
Thus a cohomology class
is the coordinate class
cp a H1(M,PL(2,C ))
of a projective structure on the topological surface M if and only if there exists a coordinate section
(za) a ro(M, 9 ((p,g')) ; the
set of all coordinate sections, module the obvious equivalence
relation, correspond to all projective structures on M with the given coordinate cohomology class.
And similarly,
(p
is the coor-
dinate class of a projective structure on the Riemann surface M if and only if there exists a coordinate section
(za) E
the set of all analytic coordinate sections, module the obvious equivalence, correspond to all projective etruatures subordinate to
the given complex analytic structure on M and with the given coordinate cohomology class.
To any matrix
T = ( as d)
projective transformation
\qt
(P(z) _
matrices. T;T'
only if
there corresponds a
= (pT a PL(2,C) , of the form
(8)
every transformation
a SL(2,C)
qp e PL(2,C)
,I b
cz+d can be so represented, and two
represent the same projective transformation if and
T' = + T .
The mapping
p: T -,> (pT
is a group homomor-
phism, forming part of the exact sequence of groups
o .-- (+ i) -> SL(2, C) -L> PL(2, C) --> 0
(9)
where
0
stands for the trivial group.
(Since
,
PL(2,C) = SL(2,C)/(+ I),
this provides the projective linear group with the structure of a complex Lie group.)
morphism µ
For any covering
_ (Ua)
of M , the homo-
clearly induces a mapping
; it is easy to see that
µ: Cl( UL ,SL(2,C)) --> Cl( Vt, PL(2,C))
this mapping takes cocycles into cocycles and preserves equivalence
classes, hence induces a mapping
µ*: HC'(M, SL(2, C)) --0 H1(M,PL(2, C)) .
(10)
If M is a compact Riemann surface and
Lemma 26.
is the coordinate cohomology class of a pro-
rp e H1(M,PL(2,C))
jective structure on M , then there exists a cohomology class T e R1(M,SL(2,C))
such that
(za) E r0(M, ( ((p,g'))
rp = µ*(T)
is any coordinate section, then there exist
a flat complex line bundle
t
H1(M,C)
r:
and a section
such that
z. = hlo/l a
It is clear that whenever
9 e H-(M,C)
h = (hla,h2a) E r(M, 0. (tT,
Remark.
Further, if
.
))
T e H1(M,SL(2,C)) , the product
gT a H1(M,GL(2,C))
and
is well
defined. Proof.
If
qp a H1(M,PL(2,C))
is the coordinate cohomology
class of a projective structure on M , select a projective coor-
dinate covering UL = (Ua,za) (%'0) a Zl( VL,PL(2,C ))
For each element
with coordinate transition functions
which represent the cohomology class
select a matrix
Tas E SL(2,C)
such that
1PQ13
(Pm = µ(T) ; so the coordinate transitions can be written
z (p)+b
a
za(P) = cz p
(11)
for p e Ua fl U0 ,
+d
where
b/
=Cam
T
OO
d
c
a$
1
K
Note that the canonical bundle
r_ 1
Kap(p)
Select any
is defined by
(cyp)+dC,0)2
=
g-1 distinct points
for p E U1 (1 U,3
p1,p2,...,pg-1 E M , where
the genus of M ; and introduce the complex line bundle
g
is
2 S = IIi
pi where
are the point bundles of §7(c).
Write
_ iK
for some
i
complex line bundle
c(C) = 2(g-1) = c(K) , then
n ; since
so we can suppose that
n e
(M,C) .
c(i) -0
Now there is an analytic
section g = (ge) E r(M, ai (c)) = r(M, Q1 (r)K)) such that the condition that
(g) = Ei
g be a section is just that
for p e Ua (1 Uo,
ga(p) = r6r(c'VzO(p) +
where (t) are constants representing the cohomology class g
Since the divisor of
is even, in each coordinate neighborhood
Ua we can select a well-defined branch of h2a(p) functions
h2a are analytic in
(12)
h2a(p) _ 9
for some constants hla(p) = za(p)h2a(p)
CO
g
)
.
The
Ua ; and
(ca,z,(p) + da$)h2p(p)
9a3 . .
n
for
p e Ua fl U. ,
Introduce further the analytic functions
It follows readily from (li), (12 ), and
these definitions, that
-181-
(13)
J
hla(p) = to(a,4,3h1S(p) +
for p e Uaf US . h2a(p) _ $C49 (coohlo(p) + d
The analytic functions
hla, h2a are clearly linearly independent.
So it follows immediately from (13) that the matrices
S00 _ 9OOTCO
satisfy the cocycle condition, or in other words, represent an element
S e Hl(M, GL(2, C)) ; and it is evident that
µ*(s) = p
.
Dividing all the matrices by their determinants will then yield a cohomology class in tions.
setisfying the desired condi-
Hl(M,SL(2,C))
Then, for the second part of the lemma, write the coordi-
nate transition functions for the coordinate section form (ii), where
The constants and
(To,)
Ta$)
(J
and by (13) the functions
define a flat line bundle,
(0aa)
satisfy the cocycle condition; determin6 a section in
ha - (hla,h2a)
while by construction
r(M, O
in the
T = (Tai) a FIl(M,SL(2,C)) , and repeat the above
part of the proof. since both
(za)
za(p)
=
hea(p)/h2a(p)
That concludes the proof. Theorem 20.
On a compact Riemann surface, the projective
structures subordinate to the given complex structure are determined uniquely by their coordinate cohomology classes. Proof.
Let
(UJ,za)
and
(Ua,wa)
be two projective coor-
dinate coverings having the same coordinate cohomology class
T ;
there is no loss of generality in assuming that the two coverings actually have the same coordinate transition functions (%4,3) a Z1(UL ,PL(2,C))
class
T a H1(M,SL(2,C))
.
Applying Lemma 26, select a cohomology such that
p*(T) = q , and suppose that
(TaO) a Z1( Vt ,SL(2,C))
isarepresentative cocycle.
Further, select
flat complex line bundles
t = (Ea0)
complex analytic sections
g = (gla,g2a) a r(M, (¢ (gT,)) and
h=
(h1a,h2a)...e
and
n = (nay) , together with
r(M, O (nT,C )) , such that za(P) - gia(P)/g2a(P) The matrices
and wa(P) = hia(P)/hn(P) Fa(P) =
are complex analytic functions in each set Ua , and
Fa(P) =
C M Therefore the functions
nab
)
for
p E
det Fa are complex analytic in each set
Ua , and n u, ;
det Fa(p) = 9C43
that is to say, det FCC
vanishes identically, or
Theorem 11.
gla
(det Fa) a r(M, a(in)) .
and
c(9n) = 0 , either
is nowhere vanishing, by
From the proof of Lemma 26, recall that the functions
g2a for instance both vanish at
fore necessarily det Fa =- 0 vectors
det F.
Since
(g1a,gga)
and
.
points; and there-
This condition means that the
(hla,hha)
pendent, or equivalently that
g-l
are everywhere linearly de-
za = wa everywhere.
Therefore the
two projective coordinate coverings coincide, and the desired result has been'demonstrated.
This theorem shows that the mapping which essociates to the projective structures on a Riemann surface their coordinate
-18a-
cohomology classes is one-to-one; hence the coordinate classes can
be used to describe the set of projective structures on a given Riemann surface.
This result is definitely false for projective
structures on a topological surface; a construction of L. Bers ("Simultaneous uniformization," Bull. Amer. Math. Soc. 66(1960),
pp. 94-97) provides coordinate classes with two different projective structures, such that the underlying complex analytic structures There
can be any two arbitrary Riemann surfaces of the same genus.
now remains the problem of determining the subset of
H1(M,PL(2,C))
consisting of the coordinate cohomology classes of the possible Before approaching
projective structures on a Riemann surface M .
this problem, as a slight digression we shall consider another description of the cohomology sets
.
for any grc+.zp
H1(M,G)
The cohomology sets
G can be
Again consider an open covering
described as follows.
of the space M .
Hl(M,PL(2,C))
= [U(X)
A chain of the covering Vt based at o E
is a finite sequence
and Ua
such that Ua = Uo
a
i
for
n Ua
i-1
0
Ua
of elements
7 = (Ua ,U ,...)Ua ) o al m
i = 1,...,m ; and
i
the chain is said to be closed if Ua = U0
also.
A simple jerk
m on such a chain consists either in replacing a pair consecutive elements of the chain by a triple
when Ua n v0 n va i
U. ,U
U011, UJ,
i
of
i+l
aU
i+l
or in performing the inverse operation.
i+l
Two chains are called homotopic if it is possible to pass from one to another by a finite sequence of simple jerks; this is clearly an equivalence relation, and the set of equivalence classes will
-184-
If
be denoted by 71'1( t )
and Oce
(Ua , ... , Ucc ) o m
, ... , Ua) are n
o
closed chains, their product is defined to be the closed chain it is clear that this product can be
(Ua I...,Ua ,Ua I...IUcc o m o n
carried over to the set chains, and that
7r1(
of equivalence classes of closed
)
(The associativity
is then a group.
7r1( Vt,)
property is obvious; the identity is the equivalence class represented by the closed chain
and the inverse of the equiva-
(Uo) ,
lence class represented by a chain
(Ua ,...,Ua )
by the chain
,...,Ua ) (Ua ,Ua o m m-1
Uo , and the notation
7r1(t/L Uo)
necessary to specify the base point.
with refining mapping such that
o
µ:
)
µ7o = U0 , then
will be used when it is
If IC is a refinement of V1,
and if there is a set o e µ
induces a homomorphism
(For any closed chain
7r1(Vt , a)
µ*: 7r1(1( ,Vo) (Va ,V
Details will be left to the
The group depends of course on the choice of the base
reader.)
point
.
is represented
m
0
I.... Va) based at o set µ(V, ,V a,...,Va
a7
o
m
1
m
it is evident that this mapping preserves
_ (µVa ,µVa,...,µVa 7.
equivalence classes, and defines the desired homomorphism µ* .) Moreover, if
V: U -> V( is another such mapping, then
µ* = V*
(To see this, it is sufficient to observe that for any closed chain (Va o
based at
...,VCC )
V.
and any index
r = 1,2,...,m-1 , it is
m
possible to pass from the chain to the chain
VVa ,..., wa (µVa ,...) µVa o r-1 r m
(PV, ,...,µVa , wa ,..., wa ) o r r-1 m
of simple jerks
(µVa
r-1
by the succession
,wa ) -,> (µVa ,µVa ,wa ) r r-1 r r
and
(µVa ,VVa
r
r
) --> (µVa ,VVa
VVC,
r
r+1
r+l
.) Now select a fixed point
)
p e M , and consider the family consisting of open coverings V( _ (Ua)
together with a fixed element
Uo a V( such that p e a.
This is a directed set, defining (U IV ) < (1( Uo) 0
refinement of Vt with a refining mapping µ(Vo) = Uo .
-A
µ:
if Y is a so that
Then put
lr1(M, p) = inv.lim. (u
7r1('U( , Uo)
Uo)
(Recall that the inverse limit group is the subgroup of the direct
product
(Y
consisting of those elements
II(Trl(U( ,U0))
U) a R(7rI(Vt ,U0)) such that µ*(y,V ,
24 'o
ever (T Vo) < (WL ,U0) .)
o) = Y o V
when-
It should be remarked that a change
of the base point has the effect of an inner automorphism on the fundamental group
7r1(M)
.
Now for any abstract group set of homomorphisms from 7r1(M,p)
ments
and
X e Hom(7r1(M,p),G)
G , let into
G .
To any pear of ele-
g e G there is associated another
group homomorphism Xg a Hom(7r1(M,p),G) defined by Xg(7r)
for 7r e 7r1(M,p) i two elements equivalent if
X = Xg for some
be the
Hom(7r1(M,p),G)
X,X a Hom(7r1(M,p),G) g ;
-
g-1X(7r)g
are called
and the set of equivalence
classes will be denoted by
Hom(7r1(M,p),G)IG Actually of course, the mapping X - Xg exiibits of operators on
G as a group
Hom(7r1(M,p),G) , and the above set of equivalence
classes is merely the quotient space under this group action. Lemma 27.
For any surface M and any group
G , there is
a natural one-to-one correspondence between the cohomology set
H1(M,G)
and the set
Hom(7r1(M,p),G)IG .
Let V _ (Ua)
Proof.
base point
containing p .
Uo
For any cohomology class
select a representative cocycle (%f3) e Zl(jtt , G) ;
q) a H1(V(, G)
and for any chain (Ua ,u
,.. .,u
,...,Ua ) _ 9a
gs(Ua ,U
0
al
fl U n Ua
u01
m-1 m
`Paa
4 0
, then %
=
i i+l
i+l
i
i.
a
9,
oal ai.
Si
is a cocycle, this last expression is unchanged under
siMPle jerks; for if
= cpa
based at U. define
)
m
0
Since c
be an open covering of M , with
Thus (A) defines a mapping
.
i+1
Xg: Tr1( Vt ,Uo) -> G , which is readily seen to be a group homomorphism. ($aa)
If
((ua)
are equivalent cocycles, then there are elements ea(PC4e0l
such that *aa =
*(Ua )U
(15)
o`x1
since Ua = Ua o m
.
and
ea e G
; so for any closed chain
,...,UCC
= ea (D(Ua ,U o
o
m
Hence, considering X
al
,...,Ua m
o
as ap element of
Hom(7r1()/t,U0),G)/G , that element is independent of the choice of (qua)
cocycle
representing the cohomology class
qP --> XT thus takes
H1(11.
G)
cp
; and the map
into Hom(7r1( Vt ,U,), G)/G .
We
shall actually show that this mapping is a one-to-one correspondence. First, suppose that Z1(V(
G)
and
(q)aa)
(4r
)
are two cocyclea in
leading to the same element in
recalling (15), it is clear that
*aa
equivalent cocycle such that
-187-
Hom(7r1(V ,U%),G) ;
can be replaced by an
*(Ua ,u ,..,ua) = g7(Ua )U o
m
a1
for every closed chain.
where element ira
8a
m
and put 0,, = *(ial)(P(7ra)
,Ua)
m-1
a7.
indicates the chain
7ral
,...,Ua )
al
Then for any element Ua a Vl. select a
chain Ira = (Ua ,U , ...,Ua o
o
in the reverse order.
Ira
The
is independent of the choice of chain Ira ; for if
is another such chain, then
a) _ (
7rr l is a closed chain. _ *(Uceua
M_1
,...,U
a]
Wa,V _
Then
,Ua ,Ua ,Ua ,...:Ua 1 m0 0
Pl = B
=
, since
al)4P(r
(Dap'G0
The cocycles
.
are hence equivalent, so the map
*(7ra)*(ir)
cp -3 XT
(cps) and
(fir
)
is one-to-one into.
Next, for any element in Hom(7r1(X ,U.),G)/G , select a represen-
tative
X e Hom(7rl( UL ,U0),G) . Again for any element Ua e Vt
select a chain
U. n UP j 0
Ua) ; and whenever Ira = NO ,U ,...,Ua o al m-1
define % = X(7ra,Ua,Up,7rp_1) , noting that
(Toe Ua,U,7rl)
Then whenever Ua n U. n y # 0,
is a closed chain.
observe that TaJfy = X(7ra,UWUy77 yl) _ rpay , since the chains (7ra,Ua,Uo,7r;1,7rp,UpU,,7ryl) and
(7ra,UaU7,,7ryl)
cycle in
are homotopic.
Thus
Z1(1j1 G) ; and it is clear that
ciated to the cohomology, class of
cp
((poo)
X
determines a co-
is the element asso-
under the correspondence
considered here, thus showing that the map
q -> Xq,
is onto as
well.
Thus for any covering lit
with a selected base point Uo a VI
there has been defined a natural one-to-one correspondence
H1(?j ,G) 4-- Hom(7r1( U(,, U0 ),G)/G .
It is evident that these
correspondences behave suitably under refinements of the covering, and hence in the limit lead to the desired correspondence; details again will be left to the reader.
The elements of H1(M,G)
Remark.
can be viewed as fibre
bundles with totally disconnected group, (flat fibre bundles); and Lemma 27 is just the classification theorem for such bundles in terms of their characteristic classes.
For a discussion from this
point of view, see N. E. Steenrod, The Topology of Fibre Bundles, §13, (Princeton University Press, 1951).
At this stage it would perhaps be of interest to see some examples.
It should be remarked that the group
7r1(M,p)
defined
above is actually isomorphic to the fundamental group of the surface, hence will be assumed known to the reader. logical appendix for some further discussion.) trivial example, the projective lint'=7P
group, 7r1(P) = Lemma 27 that
1 .
For any group
H1(M,G)
First, as a rather
has a trivial fundamental
it then follows from
consists of a single element; and consid-
ering in particular the group
G = PL(2,(), there is by Theorem 20
a unique projective structure on surprising.
G
(See the topo-
IP, which should not be very
Next, for a compact surface of genus 1 the fundamental
group is a free abelian group on two generators:
Letting 71,72
be two generators, an element
is completely determined by the values
Calling two pairs
X1X2 = X2 X1 ; thus
G X GIX.X2 = X2 X1)
Hom(7r1(M),G)
(Xl,X2)
X E Hom(7r1(M),G)
Xi = X(7ri) , and these are
arbitrary subject only to the condition that
we can identify
7r1(M) _ z + Z .
and
()j,XX)
_189-
.
equivalent if for some
element
g e G they satisfy Xi = g lXig
for
i= 1,2 , the set of
these equivalence classes can be identified with For the special case
Hom('7r1(M),G)/G
G = PL(2,C) , recall from the exact
sequence (9) that an element X E G can be represented uniquely up to sign by a matrix
T e SL(2,C)
Recall also that by an inner
.
automorphism any element T e SL(2,C)
can be reduced to one of the
following canonical forms:
(i)
/1
0
0
1
T=t
)
(ii)
;
(16)
T=\
(iii) and except for replacing
=(1
b)
0
1
T
,b#0;
a 0 0 l/a/
a by 1/a
in (iii), none of these
matrices is equivalent to. any other under an inner automorphism.
Now consider a pair of elements X1X2 = X2X1 ; if then
(X1,X2),
is a matrix in
Ti
T1T2 = + T2T1 .
Xi a PL(2,C) , such that
SL(2,C)
representing
By an inner automorphism,
T1
Xi ,
can be reduced
to one of the forms (16); thus there are three cases to consider. (i)
If T1
is the identity, then T2
is arbitrary; and by a
further inner automorphism,
T2
standard forms (16).
If T1 has the form (ii) and
T2
=\a2
b2/ d2
C2
If the
+
(ii)
, then
T2
can easo be reduced to one of the
must satisfy the condition
(a2+b1c2
b2+b1d2
c2
d2
= +
a2
a,bl+b2)
c2
c2bi+d,,
sign holds, it is easy to see that
a2 = d2 = + 1 ; while the
c2 = 0
and
- sign can clearly never occur.
The forms of
and
T2
cannot be further changed by an inner
(iii)
If
T1 has the form (iii), then
T1
automorphism.
T2
must
satisfy the condition 1,281
=+
Cc2/al b2a1
d2/al If the
+
-
a2a1 \c2 a1
b2/a1l d2/al
sign holds, it is easy to see that b2 = c2 = 0 , and
1,2 - 1/d2 ; further inner automorphisms can only have the effect of simultaneously replacing
al, a2
sign holds, it follows readily that
by 1/al, 1/a2 .
If the -
a2 = d2 = 0 , and
al = + i,
c2 = - 1/b2 ; by an inner automorphism, it is further possible to make
b2 = +1 .
Now considering all three cases together, and
writing the elements as projective transformations, the elements of
can be represented
B.1(M,PL(2,C)) = Hom(i1(M),PL(2,C))/PL(2,C)
by the distinct pairs of transformations on the following list:
Tlz = alz+bl, T2z = a2z+b2 ,
(i)
where
(17)
Tlz = - z,
(ii)
ala2 # 0,
(aZ 1)bl = (a2-1)b2 = 0 ;
T2z = - 1/z .
These pairs of transformations are inequivalent, with the exceptions that
T
T2z = az and Tlz =
z,
1
a
T2z = 1 z 1,2
are equivalent.
Two aspects of this description should be pointed out. First, the image
µ"'I'(M,SL(2,C)) C H1(M,PL(2,C))
under the map-
ping (10) consists precisely of the elements representable in the
form (17 (i)); for only in the case (17 (ii)) was the matrix
equation T1T2 = - T2T1
solvable.
This example thus shows that
the first assertion in Lemma 26 is a non-vacuous one.
By that
lemma, the element (17 (ii)) can never be the coordinate coho-
mology class of a projective structure.
Second, all the elements
(17 (i)) are actually affine, hence the cohomology classes so represented can be reduced to affine cohomology classes.
Thus,
recalling the first comment above and applying Theorem 20, it follows that any projective structure on a compact Riemann surface of genus one can be reduced to an affine structure; in other words,
projective and affine structures coincide in this case.
(c)
The main problem here is that of determining the pro-
jective structures on a compact Riemann surface in a sufficiently
explicit manner; in view of Theorem 20 and the subsequent discussion, this problem can be rephrased as that of determining
explicitly which elements of H1(M,PL(2,C)) = Hom(rrl(M),G)/G
are
the coordinate cohomology classes of projective structures on the
Riemann surface M .
Although some further discussion of
properties of complex vector bundles seems necessary before directly tackling this problem, there are some interesting preliminary results which sould be mentioned here.
The projective
structures on M correspond to projective connections, as in Theorem 19; so the question arises, how to determine the coordinate cohomology class of the projective structure corresponding
to a given projective connection on M
k a R1(M, Q-)
To begin, select a complex line bundle such that
X2
= K , where
Riemann surface.
K
is the canonical bundle of the K
Since the Chern class of
is even, and the
group of line bundles of Chern class zero has the simple form described in §8, it is evident that there exists such a line bundle.
There is not a unique such bundle of course, and for
present purposes mw choice will suffice. these bundles
It should be noted that
X can be described very simply in terms of any
projective structure on the Riem1ann surface M . choose a cohomology class
projective structure; if
covering for the structure, and
is a projective coordinate (Tcc) s Z1(Vt.,SL(2,C))
matrices representing the cohomology class dinate functions
(za)
representing that
T e H1(M,SL(2,C)) (Ua,za)
As in Lemma 26,
are
T , then the coor-
satisfy equation (11).
It follows readily
that the functions
XCO (p) - c,,z8(p)+dM ,
(18)
represent a complex line bundle
X e H1(M, m*)
p s Ua n Ud , for which
X2 = K ,
as desired. Lemma 28.
Let M be a compact Riemaan surface,
h = (ha) a C°(7/Z, O( X2)) surface, and A2 = K
X e H1(M,
(9-*)
be a projective connection on the be a complex line bundle such that
is the canonical bundle of the surface.
In each coor-
dinate neighborhood Ua of a coordinate covering of the surface select two linearly independent analytic functions
Pla(z(x)'
f2a(za) which are solutions of the differential equation
(19)
2fa(za) + ha(za)fa(za) = 0
and introduce the vector-valued functions
;
f (z ) cx
a
(flc'(Zcd 2a(zd)
Then to each intersection Ua n U0 there corresponds a unique matrix Tao a C3L(2,,C)
(20)
such that
f (za(P)) _ %C'P(P)-1T
These matrices form a cocycle µ*(T) a H1(M,PL(2,C))
(zP(P)) for p e Ua n u, .
T e Zl(lfl ,GL(2,C))
such that
is the coordinate cohomology class of the
projective structure on M corresponding to the projective
connection h . Proof.
Note that (19) is a linear differential equation
with complex analytic coefficients, and the coefficient of the highest term is nowhere zero.
Hence, as is well known, there
exist complex analytic solutions in a smell enough neighborhood
Ua of any point on the surface; and the set of all these solutions form a two-dimensional complex vector space.
If
fa(za)
is any
solution in the open set V. = za(Ua) C C , and if Ua fl UP 10 then introduce the complex analytic function
g0(z
in
zP(Ua fl U0) C VP defined by
gp(zP(P)) = ?ap(P)fa(za(P)) ,
for p E Ua fl U.
Applying the chain rule for differentiation,
gh(z.) = dz [X
(zO)fa(za)]
_ C43(z0)fa(za) + XC43(zo) lfa(za)
1911
since dz/dzo = Kcio(zp) 1 = 1
Differentiating again,
.
(zP)fa(za) +
6, (zP) P Since
(zo)-2
C43
is by assumption a solution of the differential
fa(za)
equation (19), it follows that ha(za)X8(zO)-3fa(za)
g'(zo) _
" (zO)fa(za) -
X? ,,(z.) - 2 ha(za)X,43(zI3) 4]g0(zP)
(
For the coordinate transition functions
it follows
za = fao(zo)
readily from formula (A) that
(zd ;
Q2a (z0) = 62f0(zP) _
and recalling the defining property of a projective connection as given in formula (7),
hP(zP) _ Ooneequently function in the set
(zP)-4ha(za) - A %CO(zp)"17`oO(zp)
g"0(zo)
g,(z,)
ho(zo)go(zo) ; that is to say, the
is a solution of the differential equation (19)
z0(Ua fl u0) CYO .
Now if
f1a(za)
and f2a(za)
two linearly independent solutions of (19) in the set the functions
g10(z0) - 7
fia(z(,)
and
are
za(U(x)
g2,(zB) = ?,, f2a(z(,)
are linearly independent solutions of (19) in the set zo(Ua fl u.) ; so for any other linearly independent solutions f1 (z13), f213(zO)
of (19) in
zP(UP) , the functions
unique linear combinations of the functions
gio(z0)
fi13 (zP)
This
demonstrates equation (20), and at the same time shows that the
matrices
(TC43)
form a cocycle T E Z1(t1
-1199
, GL(2,
C )) .
are
-»- ..a.a wu A VLLlu
wa(za) = fla(za)1f2(za) za(Ua) C C
.
L LA.UCGJ.Ons
in each coordinate neighborhood
Since the functions
fia(za)
are linearly inde-
pendent solutions of (19), their Wronskian fla(za)f2a(za) - fla(za)f2a(za)
is nowhere vanishing in
but this clearly means that wa(za) that function, and wa(za)
from
.
(l/wa(za))' # 0
into
0
at the regular points of
at the poles of the function
za -- wa(za)
Thus the mapping
za(Ua)
9
za(Ua)
is a local homeomorphism
IP , and the composite functions W. o za
a complex analytic coordinate mapping on the Riemann surface M
Differentiating the function wa , note that
war f la
wa =
-
ft
2a 2a
la Since the functions
fia
/
are solutions of (19),
l!/f (:'fia)--2 ha_\fia/ t
112
therefore
f
2
-(<'a)
f,
2
+\ f2a/
f,
f,
fia
f2a
la _
+ VC,
2
2a
}
f2a
=-2a f2a Differentiating once again, 2
wa =hawa+6wa( Vi`
.
2a
Finally, substituting these results into (2), it follows that
are
Recalling Theorem 19, this means that the coordinate mappings are a projective coordinate covering representing the
wa a za
projective structure corresponding to the connection.
The coor-
dinate cohomology class is therefore defined by the coordinate transition functions for this coordinate covering; and by (20), µ*
this class is just
(T)
µ*: H1(M,GL(2,C)) -> H1(M,PL(2,C))
, where
is the mapping associated to the natural mapping µ: GL(2,C) --> PL(2,C) . . Again let X2 = K
that
)
This then completes the proof.
e H1(M,
be any complex line bundle such
(r9*)
is the canonical bundle on M , and let
* (T.aO) a Zl(Vt,
)
Then consider
be a representative cocycle.
the holomorphic non-singular matrix valued functions in intersections
Ua G U0
Aa0 (z8) A
(21)
Aap
defined
by
d dz for
(z
z
e z (Ua n Up).
It is a straightforward matter to verify that
Aap(zP(p))-A07(z,,(p)) - ACer (z,,(P))
(22)
the details will be left to the reader. µ*: Hl(M,GL(2,C)) -y> H (M,PL(2,C))
homomorphism
µ: GL(2,C) ---.> PL(2,C)
compact Riemann surface of genus a
Also, let
be induced by the natural
Theorem 21. A cohomology class
µ*T
if p e Ua fl U0 fl U7 ;
H1(M,PL(2,C))
g > 1
T e H1(M,GL(2,C))
on a
has the property that
is the coordinate cohomology class of a
projective structure on the Riemann surface if and only if, for a
-- - -J-
VI - (ua, zaj or M and a
UuvCrJ.ug
representative cocycle
(T4) a Zl( 7/'(,GL(2,I)) , there exist
holomorphic non-singular matrix-valued functions various sets
in the
FCe
Ua such that Fa(p)Aap(p) -
If
Proof.
µ T
'00,0
(p)
p e Ua
for
n u, .
is the coordinate cohomology class of a
projective structure on the Riemann surface then by Lemma 28 there
exist holomorphic vector-valued functions
Ua , such that the entries in fa
fa In the various sets
are linearly independent
zLO
solutions of (19) in Ua ) and that
fa (za)XCO(zO) s Ta
(23)
(z,)
in Ua n U
Differentiating the above equation with respect to
dza,/dzP _(zd-2
recalling that
(24)
z0
and
, secure that
f'a(za)Xa (zO)-1 + fa(za)XL (zp) -
(zp)
Introducing the holomorphic matrix-valued functions
Fa(za) _ (fa(za), f'(za))
equations (23) and (24) can be written together in the form
Fa(za)Aa,(z,) = TOOFO(z0)
and since the entries of
fa(za)
tions of (19), their Wronakian These functions
Fa(za)
are linearly independent soludet Fa(z(x)
is nowhere vanishing.
are then of the desired form.
un the other nana, suppose tnat tnere exist no.Lomorpnic matrix-valued functions theorem, and let
F. satisfying the conditions of the
fa be the holomorphic vector-valued functions
consisting of the first column of the matrix Fa . f (za)
z-a
The functions
satisfy equation (23) then, so their derivatives neces-
sarily satisfy equation (24+); and the holomorphic matrix-valued functions TCOG0(z0)
also satisfy
Ga(z(,) = (fa(za),f'a(za))
Therefore
.
Ga(z(X)Aap(zP)=
det Ga(za) = det TCO-det GP(zP) , or in
other words, the functions
det Ga are holomorphic cross-sections
of the complex line bundle
(det Tap) ; and since
stant matrices, so that det Ga(za) __ 0
(det TCO)
det Ga(za)
or
=
are con-
has Chern class zero, either In the second
never vanishes.
case, it follows immediately that the mapping
defined by wa(za)
TCO
fia(za)/f2a(za)
,
za ---J Wa(za)
,
is a local homeomorphism;
and from equation (23), the local coordinate functions
wa.za
define a projective structure on M. with coordinate cohomology class
T .
show that
Thus to prove the theorem, it is only necessary to det Ga(za) A 0 .
Now if
in the vector-valued functions
det Ga(z(,) = 0 ,
fa(za)
would be linearly dependent;
thus these functions could be written in the form where
fa(zes)
Note that neither
anywhere, since the original function non-singular.
f (za)= fa(za)C'
are ordinary holomorphic functions and
constant vectors.
the entries
ca
are
nor S. can be zero
fa(zes)
was by assumption
Fa(zes)
Equation (21) would now have the form
fa(za)Xap(zp)Zac = fP(zP)Tap=p but this would imply that
fa(za)%043(zP) = c
fP(zP)
CAB
for some
constants
This latter condition is impossible; for since
ca13 .
the functions
fa(za)
X00
would necessarily be equivalent, but their Chern
and
c
classes are
and
g- 1
are nowhere vanishing, the line bundles
0
respectively.
This contradiction
completes the proof of the theorem. Corollary.
The functions
Fa(za)
in Theorem 21 can be so
chosen that
dz Fa(Q a where
(ba(za))
0
-4- ha(za)
1
0
Fa(z(x)
is the projective connection corresponding to the
projective structure. Proof.
The matrix function F(za)
can always be taken
in the form Fa(za) _ (fa(za),f'(za)) , where(za)
is a
solution of the differential equation 2(za) + ha(za)f'a(za) = 0 as in Lemma 28.
Then
(Za), - 2 ha(za)(za) )
dza Fa(Za) _ (,f ' (Za),(Za) ) _
=a a I za a
0
2 ha(za)
1
0
= (f (z ) f'(z ))
as desired.
Let !b 2
be the sheaf of germs of complex analytic
mappings from a Riemann surface M into the group
GL(2,C) ; note
that this is merely a sheaf of sets, with no group structure defined.
As in the discussion on page 174+ and the following pages,
it is possible to introduce a non-abelian cohomology set l(M'/d2); the elements of this cohomology set are called complex vector bundles
of rank 2 over the Riemann surface M .
The concept is straight-
forward, in view of what has been discussed so far, but a more detailed treatment will be postponed to a later section. constant functions form a subsheaf
The
GL(2,C) C b 2 , and the
inclusion mapping between these sheaves induces a mapping i
:
H1(M,GL(2,C)) -- H'(M,
2) ; thus the flat complex vector
bundles introduced on page 178 determine well-defined complex vector bundles.
In this terminology, Theorem 21 asserts that the such that
flat vector bundles
T e H1(M,GL(2,C))
µ*T s K1(M,PL(2,C))
is the coordinate cohomology class of a
projective structure on the Riemann surface are precisely those flat vector bundles such that
i*T = A E E (M, 'b 2) , where
A
is
the complex vector bundle defined by the transition functions (21). The problem is that of determining explicitly the flat forms of that canonical complex vector bundle.
(d)
On a surface endowed with a projective structure, there is
a generalization of the deRham sequence which is of some interest;
this was introduced by M. Eichler, who used it in studying algebraic correspondences on some Riemann surfaces arising in number theory.
(See M. Eichler, "Eine Verallgemeinerung der Abelschen
Integrals,"
Math. Zeitschr. 67(1957), 267-298; and G.Shimura,
"Sur les inte'grales attaches aux former automorphes," Math. Soc. of Japan 11(1959), 291-311.)
Jour.
As yet this structure
has not been studied extensively on general Riemann surfaces.
M be a Riemann surface with a fixed projective
Let
structure, and let Chern class ing
c(%) = g- 1
.
Choose a projective coordinate cover-
M , belonging to the given
za = f(z) be the coordinate
projective structure, and let transition functions; thus
be a complex line bundle of
LIL*)
of the surface
(Ua,za}
jtl_
X e H1(M,
(fap) E Z1(VC,PL(2,C))
represents
As
the coordinate cohomology class of that projective structure.
in Lemma 26, choose matrices
(Tap) a Z1( lit , SL(2, C)) which
represent these projective transformations, that is, such that * µ (Tap) = fap .
Writing these matrices out explicitly as aap
(25)
Tap =
cap
recall as above that the functions
(capzp + dap) represent a
complex line bundle of Chern class
g - 1 ; therefore it is
evident that
for p e Ua fl Up ,
(26)
XCO(p) = gap(cCOzp(P) + dap)
where .(gap)
are complex constants representing a complex line
bundle
g E 11(M,
(g) of zero Chern class. consider the sheaf 04%-n)
For any integer n > 0
of
germs of holomorphic cross-sections of the complex line bundle ,-n
.
If
p e Ua is a point on the surface and
is a germ of a cross-section at
an analytic function if
(27)
fa(za)
p , then
f
f s 64X')
p
is represented by
in some neighborhood of p ; and
p E Ua fl up , these local representations are related by
fp(zpap(zp)nfa(za)
ap(c zp+dap)nfa
(az, ca,'+dap/I +b\
Having written these conditions out so explicitly, there is an -(,-n)
obvious subsheaf of
za of degree
is a polynomial in
Note that if
is a polynomial in
z
of
Therefore we can introduce the subsheaf
also.
consisting of those germs of analytic sec-
11 n(T n) c (.(T. n)
n which are polynomials of degree < n
tions of
fa(zes)
< n , then from (27) it
f0(z0)
necessarily follows that
degree < n
as follows.
,
in any local
projective coordinate system of the given projective structure. For the case
Fo(x°) = C , the subsheaf of constant
n = 0,
functions; and this subsheaf is the kernel of the sheaf homomor-
phism
d: 6 -4 O-(K)
defined by exterior differentiation.
As
a generalization: Lemma 29.
On any Riemann surface M with a fixed
projective structure, there is an exact sequence of sheaves
for any n > 0 , where to a germ
f s
LJ
fa(zes)' the germ
(X-n)
do+l
represented by a local analytic function
p
do+lf
is the homomorphism which associates
represented by the local
a
analytic function do+lfa(za)/dza
1 do+l
Proof.
The first thing to prove is that
defined sheaf homomorphism. two local analytic functions
If
f e d-(? n)p
fa(zes)
and
fP(z
is a well-
is represented by
when
p e Ua n u, , those two functions are related by formula (27). Select a simple closed curve z0(p)
and such that
fP(zP)
713 C Z0(U
encircling the point
is analytic on an open neighborhood
form
where
*
6(Kn+1x-n) )
0 -+ r(M,
(29)
* Fn(,-n)) L> r(M, & (T1nKn+1x-n))*+ 0,
H1(M,
r(-)* denotes the dual vector space to
Proof.
The complex line bundle
X
r(-)
.
can be defined by a
cocycle of the form (26) for some flat line bundle
E
,
and
= g2.
The exact sequence (28) of Lemma 29 can be rewritten
0 ---> V ('1)
n) --> 0
OL Wn) do-> LIQ (Kn+l
The corresponding exact cohomology sequence on the surface
M then
begins
0 T` r (M, V
(,n))
--
62
r (M,
(,-n))
H1 (M, ?n(X n)) > Hl(M, (Q ( n)) Since the line bundle
k-31
--T
r(M, 6.
(d') r Hl(M, &(Kn+1T n)) -> ...
has Chern class
c(%-31) = -n(g-1) < 0
it follows from the Corollary to Theorem 11 that then of course theorem, and
since
r(M, T n(A n)) = 0
as well.
n (M, a (X-")) = r(M, (V (KO))*
H1(M, Lk (Kn+1X n))
.
r(M, 0
(,,-n))
= 0;
r(M, I (TnKn+l%
n))*
&(,,i, -n))*
= 0 ,
Upon substituting these results
in the exact cohomology sequence, there follows the exact sequence (29), as desired. Corollary.
If the complex line bundle
ceding theorem is such that
T.2n
X
in the pre-
= Kn , then
dim H1(M, f n(x n)) = 2 dim
,
From the Serre duality
r(M, & (K n%n))* = r(M,
c(,n,--n) = c(% n) < 0
-
n)) S*
(Kn+1
r(M,(O(Kn+l%-n))
Proof. In I
The additional hypothesis is essentially just that
= 1 ; the first and last terms in the exact sequence (29) then
have the same dimension, from which result the Corollary follows immediately.
R-(M, pn(X-n))
The groups
are called the Eichler coho-
mology groups associated to the given projective structure on the
Riemann surface M .
For the special case
n = 0 , note that
H1(M, T° (x°)) = H1(M,c) , the ordinary cohomology of the surface M .
For any value
these cohomology groups are finite-
T) > 0 ,
dimensional complex vector spaces, as a consequence of Theorem 21; and indeed, these groups can be described explicitly in a purely algebraic manner, in terms of the coordinate cohomology class of the given projective structure and the cohomology class of the flat line bundle
.
n
Recalling from part (b) above that the coordinate
cohomology class can be viewed as an element of Hom(rrl(M),PL(2,C))/PL(2,C) , and that the bundle
n
can be
described as an element of Hom(Trl(M),C) , it is evident that the description involves only the fundamental group of the surface M. There is at present no need to carry out this description in great detail, so we shall consider the matter only rather briefly. element
a e H1(M, p n(T n))
can be represented in terms of a
projective covering ul = {Uaza) pn(T.-n))
aa6 a z1( IlL, in the variable
zP
by a cocycle
; each element aa,,(z.) is a polynomial
of degree at most n , and the cocycle con-
dition is that
(30)
An
acey,(zy) _
%
(zy)n aap(fP7(zy)) + apy(zy)
of the closure of the interior of
yp ; and suppose that
selected sufficiently near
that the function
zp(p)
yp
is
fa(za)
is
analytic on an open neighborhood of the closure of the interior
of ya = fap(yp) C za(Ua) , where
za = f(z) is the coordinate
transition function. zp(Ua n up)
za(Ua n up)
4
za= fav(zp)
Applying the Cauchy integral formula, do+lfa(z a ) fa(5a)
n+1) dzn 1
2
f .
to
Now make the changes of variables where
ya (to - za)n+2
za = fap(zp),
tca = fap(sp) ,
is the projective transformation represented by the fap
unimodular matrix (25).
a00 P + bap dap cap p +
ta- za
_ (gyp -
Note that
5p
aapzp + bap
capzp + dap
cap p +
ap
zp
capzp
zPA2
and that dta = (capsp + da43)-2dtp = tap xap(5p)-2dt
and recalling (27), it follows that
;
do+lf
z
a( (X)
(n+1)
_
2'rri
dznn+l
2n-2
f P
E04 Y
fP(
%ap(z0) n+2
0
)
d
P
- z5 )n+2 6
P do+lfP(z
-2n-2
_
aP(zP)
n+2 dzn+l
P
Thus these derivatives represent the same element of so that the mapping
do+1
in
(,-2n-20+2),
is well defined.
The kernel of the homomorphism
do+l
is the subeheaf con-
sisting of those germs of analytic functions
fa(za)
such that
do+1fa(za)/dza
1 = 0 , hence is just the subsheaf
1? n(' n) C ® (,-n)
of polynomials of degree at most
n .
This
shows the exactness of the sequence (28) at the first two places, where
i
is the inclusion mapping.
exactness for any element ative analytic function function
To complete the proof of -2n-2,n+2)p
f e ((
fa(za) ; choosing any complex analytic
in an open neighborhood of p
ga(z(,)
do+lga(za)/dza 1 = fa(za) , the function element
g e (P ().-n)
part of the proof.
select a represent-
for which
dh1
Thus the mapping
ga(za)
g = f do+l
,
such that
represents an
in view of the first in (28) is onto, and
the proof is thereby concluded. Theorem 22. genus
Let M be a compact Riemann surface of
g > 1 with a fixed projective structure, and let
x E H'(M,
(9 )
be a complex line bundle of Chern class c(T.) =g-l;
thus there is a, flat line bundle
n
such that 0 = IK , where
is the canonical bundle of the surface M .
integer n > 0
K
There is then for any
an exact sequence ogcomplex vector spaces of the
whenever ua n U0 n Uy # 0 , where dinate transition function.
f,, c PL(2,C)
is the coor-
Note that (30) is a formal identity
among various polynomials, and does not involve the point set
Ua n U0 n U7
explicitly.
there are polynomials
A cocycle
Ta(za)
aa43
is a coboundary if
of degree at most n
such that
aao(z0) = T0(z0) - xap(zp)n Ta(fap(zP))
(31)
whenever Ua n U
and the cohomology group is the quotient
of the group of cocycles by the group of coboundaries. It is perhaps of some interest to see the explicit forms of the homomorphisms in the exact sequence (29).
of a projective coordinate covering f e r(M, (D(Kn+lT n))
.
= (Ua,za) , a section
is given by analytic functions
fa(za) a r(za(Ua), dl) in Ua n tr
First, in terms
such that
The mapping
8
(ZO)n+l%00(10)-nf0(Z0
fa(za) = K
is the coboundary operator derived
in the familiar manner from the exact sequence of sheaves (28).
Thus in each set
Fa(za)
za(TJaa) C C
select a complex analytic function
such that d-Fa(za)/dza l = fa(za) ; any (n+l)-fold
indefinite integral of the function mology class
The coho-
will do.
* b f s 111(M,
is represented by the cocycle
'p n(X-n))
aa43 (z0) = FP(z0) -
(32)
The functions
fa(za)
aa3(zP)
Ta(za)
.
are necessarily polynomials of degree < n
then; and replacing the functions for polynomials
XCO(2 )n Fa(za)
Fa(za)
by Fa(za) + Ta(za)
of degree < n , the most general possible
choice for these functions
Fa(za) , replaces the cocycle (32) by
a cohomologous cocycle, in view of (31). mology class
aC43 e
a e H'(M, ?n(x-n)) , represented by a cocycle
Z 1(1l(, 'n(C-n)) ; the element
on the space
Next consider a coho-
r(M,
( (gnKtt+l% n)
, described explicitly as in the
Namely, select any zero
discussion of the Serre duality theorem. cochain
g = (ga) a Co( IX ,
C"
select
functions
such that
G- (Tn))
za(Ua)
in
ga (z(,)
aa,(z.) = g,(z,) - Xbo (zo)nga(fQP(z,))
that
(ag) a z°(V
element
F O,1(, n)) = r(M,
,
is a linear functional
V*a
in
bg = a ; that is,
such that zP(Ua fl u.)
Note
.
10'1(X-n))
For any
.
f = (fa) e r(M, tj (,nKn+lX n)) = r(M, (ul'0(,In Knx-n))
the form aga ,. fadza E r(M,
e1,1(1,nKnX-2n))
= r(M,
l'1) ; and
then (v*a)(f) = ff,, aga A fadza .
(33)
* For the case
the image
n = 0 ,
f e r(M, m (K)) = r(M, (Q-1'0)
the differential form
S f e H1(M,C)
of an element
is essentially the set of periods of
f ; and as above, for any n > 0
* b f e H1(M, mn(,:n)) of an element
-n))
f e
viewed as a generalized set of periods of the element from an (n+l)-fold indefinite integral.
the image
f ,
can be
derived
For any cohomology class V*a
a e H1(M, T n(%-n))
the vanishing of the image
,
or equiva-
lently the vanishing of the integrals (33) for all elements f e r(M, CO (,,nKn+1? n))
that r(M,
a (9
,
is the necessary and sufficient condition
be the generalized periods attached to a section in (Kn+l%-n))
.
The Eichler cohomology group can also be described slightly differently.
It follows almost immediately from Lemma 29
that, on a Riemann surface M with a fixed projective structure, for any integer
n > 0
there is an exact sequence of sheaves of
a
the form
(34) where
n+l
n(,-n) _i_>
0
(g-2n-20+2)
denotes the sheaf of germs of meromorphic functions.
T h e homomorphism
do+l
in (34) is not onto, since a general mero-
morphic function does not have single-valued meromorphic indefinite integrals of all orders.
do+l M (? n)
T h e image
is called
the sheaf of germs of meromorphic sections of the second kind of the bundle
X -n ,
evident that, if
za
p e M such that
za(p) = 0 , then
.
It is
is a local coordinate mapping at a point
q-2n-2An+2
subsheaf
0q-2n-21n+2)
and will be denoted by
p
O
is in the
(,_2n-2.n+2)
f e '
p
precisely when the germ f
sented by a meromorphic function
fa(za)
is repre-
with a Laurent expansion
of the form
£
fa(za) =
(35)
E avzv
a_vz-v +
v=n+2
v=o
for these are just the meromorphic functions which can be written
fa(za) = do+lga(za)/dzn 1
for some meromorphic function
The condition that a meromorphic function
fa(za)
ga(zes)
have a Laurent
expansion of the form (35) is not invariant under an arbitrary non-singular analytic change of coordinates, considering as an element of the sheaf #
l,! o(
2n-2 n+2 T.
)
(,-In-10+1)
1,0 P
; so the sheaf
depends upon the choice of the projective
structure on the Riemann surface. tional here.
(fa(zes))
For the elements
(The case
n = 0
is excep-
(fa(za)) a (g-2%2)p = rM(K)p =
can be viewed as meromorphic differential forms on M
and condition (35) can be rephrased as the condition that the form have residue zero; this expresses the condition intrinsically, in The terminology "mero-
terms of the complex structure alone.
morphic functions of the second kind" is motivated by the usual terminology in this special case.) Theorem 23.
Let M be a compact Riemann surface with a
fixed projective structure, and let line bundle of Chern class
T. c H1(M,
c(%) = g-l
-
*)
be a. complex
Then for any integer
n > 0 , the Eichler cohomology group can be written canonically
I'(M,dn+l 7 (:n))
.
ti
(M, P n(,n))
dn+lr(M, ?y((_:-n))
Modifying (34+) to yield the exact sequence of
Proof.
sheaves
where
dn+l
V,(, -n)
0
do+l (,-n)
(T.n)
r
_ ry,, o(g-2n-21n+2) C
(,,-n) - 0
> do+1
(g-2n-2 T.n+2 )
, the asso-
ciated exact cohomology sequence begins
0 -k r(M, 11 n(T n)) (36)
-.
> r(M,1)
H1(M, 'fi
(,-n) ) dn+l > r(M,dn+1 M. (,:n))
S
...
.
n(,:n))
--> R1(M,'h, (,-n) ) -.
.
* Selecting any section
plication by g
g E r(M,
(Xn)) , the operation of multi-
defines a sheaf isomorphism ?} (,-n) _
fore, by Theorem 12,
H1(M, q (,,n)) = H1(M,11'( ) = 0
.
; there-
The desired
theorem is then an immediate consequence of the exact sequence (36), and the proof is thereby concluded.
Corollary 1.
On any compact Riemann surface M it
follows that
H1(M, c) = r(M, d Proof.
"'C
)/dr(M, 11) .
This is just the special case
recalling that in this case the sheaf
d
n = 0
C '1'0
of Theorem 23,
is defined
intrinsically, independently of a choice of projective structure on the surface. Remarks.
r(M,d
In the Corollary, the space
)
is the
space of meromorphic differentials of the second kind on the
Riemann surface M , that is to say, is the space of meromorphic differentials with zero residue at each point of the surface, The Theorem itself can be restated somewhat more precisely as the
assertion that, when the genus
g > 1 , there is an exact sequence
of the form
(37)
o - r(M,
(T-n)) dn-----> I'(M,dn+l ( n)) -
.
for as in the proof of Theorem 22 it follows that
H (M, n(x n))
o;
r(M, r n(N-n)) =0
,
so that (37) follows directly from (36). For some purposes interest lies not just in the Eichler cohomology group itself, but also in the splitting of that group given by the exact sequence (29). §8(b).)
(Compare the discussion of
Thus one is led to consider the form of that splitting
when the Eichler cohomology group is represented as in Theorem 23.
To be explicit, consider a cohomology class
o e
(M,
which is represented by a section h = (hcl) a r(M,dn+l27 under the isomorphism given in Theorem 23.
cient to describe the image of
c
n(%-"))
(,,-n))
It is clearly suffi-
under the homomorphism
v
of (29), in terms of the section
h ; recall that
linear functional on the vector space each set
v v
is a
(nnKn+l%-n))
r(M, (1
For
.
Ua of a suiteble projective coordinate covering
there will be a meromorphic function Ha(za)
such thet
do+lHa(za)/dza l = ha(zes) ; these functions can be viewed as an element
(H(X) a Co{Vt
For any section
(,-n)
,
the products
f = (fa) a r(M,
fa(za)Ha(z(x)
an element fH = (faHa) a Co( , 1'41(,nKn+l%-2n) ) ti Co(
1,0)
form
(K))
Co(Zq
=
i the residue of this meromorphic differential
form is well defined locally by the Cauchy integral formula, and its total residue on the surface will be denoted by
If M
Corollary 2. genus
g > 1
and
P,._(fH)
.
is a compact Riemann surface of
a e H1(M, `r n(, n))
is represented by
(dn+1lia)
h = (ha) =
a
r(M,dn+l V(,. n))
under the isomorphism of
* Theorem 23, then under the homomorphism image
of Theorem 22 the
v
is the linear functional whose value on an element
v*(o)
f = (fa) E r(M,
LV
(,nKn+l%
is given by
n))
27ri k(m) Proof.
If
(aCO) a zl(
(ha) = (dn+
a)
v
is a cocycle repre-
be represented by the section
( n))
a r(M,do+1
by Theorem 23 is just that
(38)
P n(T n))
a , then it is readily verified that
senting the cohomology class the condition that
,
under the isomorphism given
l
vaO(zo) = H,(z.)
Ha(zes)
-
Suppose that the projective coordinate covering so chosen that the poles of the functions
Ha(za)
t = (Uaza) are each
is
contained in only a single coordinate neighborhood. the functions
Ha(za)
by
CO*
Multiplying
functions which differ from one
only in small neighborhoods of these poles, and which vanish
identically in some neighborhoods of these poles, yields e functions
ga(za)
section Ua fl U0 .
mapping
V
which also satisfy equation (38) in each interRecalling the explicit description of the
, as in (33), it follows that
V*(a)-f = 1 M aga . fadza The latter integral will vanish identically except for those
neighborhoods
Ua containing poles of the functions
for outside of these neighborhoods,
ga(za) = Ha(za)
Ha(za) is holo-
For such a neighborhood U. , however,
morphic.
tt aga
Ua
faaza = t! N(fagadza) = tt d(fagadza)
Ua
Ua
faHadza = dUa fagadza
= 2Tri
of
1(fH)
which yields the desired result.
(e)
The projective structures and their associated coordinate
cohomology classes can be given a geometrically appealing global
N formulation as follows.
Let M be the universal covering space
ti ` of a surface M , with covering mapping Trr(M)
IT: M T M , and let
be the fundamental group of the surface M .
familiar,
Trl(M)
As is
can be viewed as a group of homeomorphisms of
ti
M onto itself, commuting with the covering mapping
that
M/7r1(M) = M .
(Identifying the group
7r1(M)
IT
and such
as defined in
(b) above with the usual fundamental group, this interpretation can be found for instance in Seifert-Threlfall, Lehrbuch der Topologie, chapter 8 (Chelsea, 1947).
Alternatively, this result
can be derived directly from the discussion in (b), parodying in simplicial terms the standard construction.)
If M has a pro-
jective structure, it induces a unique projective structure on M
A by means of the mapping
7r: M -k M .
For if
(U(X,za)
projective coordinate covering of M such that the sets
is a
Ua
are
connected and simply-connected, then each connected component of
7r l(Ua) C M will be homeomorphic to Ua under the mapping and the functions
7r
,
zaoir can thus be used as coordinate mappings
on each such component; it is evident that this is a projective ti
coordinate covering of M , and that equivalent projective coor-
dinate coverings of M induce equivalent projective coordinate coverings of M .
Note that the mappings in
7r1(M)
are Projec-
tive transformations of M for the given projective structure, in the sense that they are represented by projective functions in
terms of local coordinates for any projective coordinate covering belonging to that structure; this is quite obvious, since indeed the mappings are represented by the identity functions in the above coordinate covering. Now since
7r1(M) = 1 ,
it follows from Lemma 27 that the ti
coordinate cohomology class of any projective structure on M trivial; hence there is a projective coordinate covering
is
(Ua,z a)
ti
of M representing the given projective structure and such that
the coordinate transition functions mappings.
The various coordinate mappings
global mapping
mapping
p
are identity
za = fC413 (z0)
za then define a
p: M --k D from M onto a subset D C P P.
is a local homeomorphism, so the image D
nected open subset of the projective line
]P
.
The
is a con-
Note that any
other such coordinate covering representing the same projective
structure will define a mapping
N ` pl: M T Dl
also, but
pl =Rop
for some projective transformation R ; to this extent the mapping p
is determined uniquely by the projective structure.
T e 7r1(M)
If
is a covering translation mapping, then T will be
represented by a projective transformation in terms of any local
coordinates for the given projective structure of M ; thus for
any point po e M there will be an element that
for all points p near po , since
p(Tp) = T(pp)
defined by the local coordinate mappings.
such is
p
Clearly the element
is independent of the point po e M ; for the corre-
T e PL(2,C) spondence
T e PL(2,C )
po -k T
is locally constant as noted above, and M
is connected.
Therefore for any transformation T e 7rl(M)
is an element
T e PL(2,C)
such that
p(T p) = Tp(p)
(39)
there
It is evident that each T
for all p E M
necessarily maps the domain D onto
itself; and that the mapping
per:
p*(T) = T , where
are related by (39), is a group
homomorphism.
T
and
T
Note that if
pi = Rop
PL(2,C)
defined by
is another such mapping
representing the same projective structure, then p1() _ = Rp(T p) = RTp(p) = RTR-lpl(p) ; hence
pl = Rp R
1
.
This pair
of mappings
p: M -4 D ,
(40)
p*: Ir1(M)
-4 PL(2,C)
related by
for all
p(T p) =
(41)
T E 1T1(M) and p E M
will be called a geometric realization of the given projective Note that the mapping
structure on M .
local homeomorphism, and that geometric realizations
(p,p*)
p
is a complex analytic
is a group homomorphism.
p
and
(p1,p1)
will be called
equivalent if there is an element R e PL(2,C) and
PI = Rp R
1
.
Two
such that
p1 = Rp
The previous observations show that there is
a natural one-to-one correspondence between projective structures
on a Riemann surface M and equivalence classes of geometric realizations; for it is apparent that any geometrical realization
determines a projective structure on the surface M . (p,p*)
If
is the geometric realization of a projective
structure on M , the mappings
p* belonging to all equivalent
geometric realizations form an element PL(2,C)
.
(p*) a Hom(7r1(M),PL(2,C))/
It is easy to see that this element is precisely the
image of the coordinate cohomology class of the given projective structure under the homomorphism of Lemma 27; the verification is straightforward, and will be left to the reader.
This provides
the most convenient way of looking at the coordinate cohomology classes of projective structures on a Riemann surface.
As an example, consider the analogous construction for
affine structures; thus, let M be a compact Riemann surface of ti
genus
1 with a fixed affine structure, and let
p: M -> D
and
P
:
7r1(M) --- A be the geometrical realization of that affine
structure, where
D
is a subdomain of the complex line and A
the group of affine transformations.
Here of course
is a
N
N free abelian group on two generators
7rl(M)
is
and
T1
T2 ; and recalling
the discussion of equation (17), the homomorphism
p* will have
N one of the following forms, where
are the indicated
Ti = p*(Ti)
affine transformations:
Tlz = z + b1, (42)
I
Tlw = alw,
(ii)
T2 z = z + b2
* For the quotient space
ala2 # 0
T2w = a2w,
D/p 7r1(M)
in case (i) it is necessary that
to be compact, it is clear that b1
and b2
be linearly inde-
pendent over the reals, and D = C ; and in case (ii) it is necessary that
D = C
I aiI / 1
for either
i = 1
is an isomorphism.
p
is a homeo-
Thus we can identify M = C,
rri(M) = (gmap of translations generated by T1
and M = C/7r1(M) .
i = 2 , and
p: M - D
In case (i) it is evident that
morphism and
or
and
T2
in (i)),
This is just the familiar representation of a
compact complex torus, as discussed in §1(f), and provides the simplest affine structure on the torus.
p: M -> C
cations, consider the mapping w = p(z) = eCz
for a complex constant
covering mapping, exhibiting of
* C
.
Retaining these identifi-
p
c
defined by 0 .
This is a
as the universal covering space
Furthermore,
P(Tiz) = p(z +b i ) = ec(z+bi) = aip(z) = Tip(z) ,
cb i where
and Ti
ai = e
Thus evidently
p
*
are the transformations given in 42(ii).
and the homomorphism pTi T T. form a
geometric realization of another affine structure on the same underlying Riemann surface.
c = 0 , and these additional affine structures for
for the case all values
(Note that the affine structure 42(i)
c # 0 , are easily seen to be distinct, and to be all
the affine structures on that Riemann surface.
Thus, as noted in
part (a), the set of all affine structures on a given complex torus are in one-to-one correspondence with the complex numbers c e C .)
These examples all have the property that
p: M -3 D
is not just a local homeomorphism, but a covering mapping. group
p 7r1(M)
The
does not always act in a discontinuous manner on
* the domain D = C
;
so although there is always a continuous
mapping M = M/'rrl(M) - D/pirl(M) induced by p: M ---> D ,
it is
not necessarily a homeomorphism, nor even a covering mapping.
It
may be observed that the same coordinate cohomology class (or equivalently the same group
p*7r1(M)) can be associated to affine
structures on inequivalent complex tori; thus Theorem 20 definitely requires consideration of a fixed underlying complex structure. For some further discussion of this geometric realization, see R. C. Gunning, Special coordinate coverings of Riemann surfaces, Math. Annalen, 1966.
(After this had been written there came to my attention the following paper, which contains some related results: Hawley and M. Schiffer, Half-order differentials on Riemann surfaces, Acta Math. 115 (1966), 199-236.)
N. S.
§10.
Representations of Riemann surfaces.
(a)
Perhaps the simplest concrete representation of a Riemann
surface is as a branched covering of the projective line
To
]P.
describe the general topological situation, consider two 2-dimen-
sional manifolds M and N .
A continuous mapping
is called a local branched covering if each point
open neighborhood U C M such that restriction of f of
exhibits
f(U)
f: M -> N
p e M has an
is open in M , and the
as an m-sheeted covering space
U-p
for some integer m ; the integer m-1 will be
f(U) - f(p)
called the branching order of the mapping
and will be denoted by of(p) . point of
f
if
of(p) > 0
.
The expression
The point
at the point p ,
f p
is called a regular
of(p) = 0 , and a branch point of
M of(p)
Ep
f
if
is called the total
E
branching order of the mapping f .
Note that the branch points
form a discrete subset of M ; the mapping and
f
is an open mapping;
f
is a local homeomorphism in a neighborhood of any regular
point.
If M and N are Riemann surfaces, then any non-trivial
complex' analytic mapping
f: M -> N
the branching order at a point derivative of that is,
f
p
is a local branched covering;
is the order of the zero of the
in any local coordinate systems in M and N
of(p) = vp(f') .
(For a discussion of the topological
properties of analytic mappings, see for example L. Ahlfors, Complex Analysis, pp. 130-133, (McGraw-Hill, 1966).)
verse assertion in some sense, if
f: M -k N
As a con-
is a local branched
covering between two topological manifolds and N has a complex structure, then M has a unique complex structure for which
f
The proof is straightforward, and will be
is an analytic mapping. left to the reader.
A
The global form of this situation is also of interest.
f: M -> N between two topological surfaces
continuous mapping
is called an r-sheeted branched covering if covering and if for every point
Z
(of(P)+1)=r
p1,p2,...,ps e M be the points of
Fixing q E N , let
and select open neighborhoods local branching of
f
at
Ui
around pi
pi ; the sets
enough to have the same image under
f(Ui)- q
is a local branched
q e N ,
(p a MIf(p) = q)
then exhibits
f
Ui(Ui - pi)
f .
exhibiting the can be chosen small'
Ui
The restriction of
f
as an r-sheeted covering space of
It is thus evident that the image under
.
f-l(q)
branch points form a discrete subset
f
of the
(g1,g2.... ) C N ; and that
f: M - Uif 1(qi) --> N - Uigi
is an r-sheeted covering space in the ordinary sense. Theorem 24+.
Let M be a compact Riemann surface of
genus
g , and
E e H1(M, B *)
class
c(E) = r
.
f0f1 a r(M, B'(g))
be a complex line bundle of Chern
then to any pair of complex analytic sections
which have no common zero on M there is
canonically associated a complex analytic mapping
f = (fC,f1): M - IP
,
which is an r-sheeted branched covering with total branching order
2(g+r- 1)
.
Proof.
Let Vt,= (U.,za)
be a complex analytic coordinate
covering of M , and (ga4) a Z'( jf, dl*) be a cocycle representing the line bundle
g
complex analytic functions = EaO(p)fip(z0(p))
The sections
.
are represented by
fi
fia(za(p)) _
fia(za) , such that
The map from Ua to
whenever p e Ua fl U0 .
IP , defined in terms of homogeneous coordinates on P by P
--> (foci (za(p)),fla(za(p))) e P , is clearly complex analytic;
and the two maps thus defined in Ua fl UP agree, since and
(Etof0p,ltoflp) _ (f0a,fla)
(f0J3,flP)
represent the same point in P.
This then defines a complex analytic mapping
f: M --> IP , which
exhibits M as a local branched covering of
IP.
points
a = (a0,a1) a IP
precisely when such that
and p e M , observe that
f(pi) = a
local coordinates in P
near
Furthermore,
f has the local description
the mapping
- fla za
pi ;
.
0 , then in terms of inhomogeneous
f0a(za(P))
P
pi e M
are precisely the zeros of the complex
if pi a U. and if say al
p
f(p) = a
alf0(p) - a0f1(p) = 0 ; thus the points
analytic section h = alf0 - a0f1 a T(M, (¢ (9))
for
Next, for any
P
so
ha(za(P))
al + aifla za
p
consequently
o f(Pi) = vpi(foo/fla)' = vpi(ha/alfla) -1 = vpi(ha) - 1 since fla(pi)
0 . From Theorem 11 it then follows that r = c(g) = Z v pi (h) = L (of(pi)+ 1) ; i
so that
f: M - IP
is actually an r-sheeted branched covering.
Finally, introduce the analytic functions
fOct (za)
f'0a(za)
f]a(z(X)
fia(za)
ga(zes) = det
in the various neighborhoods
za(Ua)
Since in terms of inhomo-
.
geneous local coordinates the mapping
f
p - fOa(za(p))/fla(za(p))
fl(p) / 0 , it follows that
wherever
1a) = vp(ga) ; and the same result
of(p) = vp(f0/fla)t = vp(holds at those points where
has the local description
f0(p)
0 .
The total branching
order is then just the total order of the functions
ga
Note
that for points p e Ua fl u, , V za(p)) = dza 19CO(ZO(p))'fiO(zO(p))3
p(zp(p)) + 9
=
(zo(p))'fia(z0(p))),
dz
where
K., = dO define the canonical bundle of M ; it readily
follows from this that
ga(za(p)) =
(ga) E r(M, (y-(K92))
Applying Theorem 11 once again, the total
branching order
b =
b
.
(p)g,(z,(p))
,
so that
is
Z of(p) = vp(ga) = c(Kg2) = 2(r+g-1) E P e M p e M
thus completing the proof. Now suppose that
f
is a non-constant meromorphic function
on a compact Riemann surface M ; as in §1(e) that function can be considered as, an analytic mapping
divisor of the function
,,¢ (f)
r
i=1
f
f: M --> IP
.
Note that the
can be written in the form
where
pi # qj
,
for from Theorem 1.1
the total order of the divisor of f must be zero; the integer r,
the total order of the zeros of f , will be called the degree of that function.
Corollary 1.
of degree
r
If
f
is a non-constant meromorphic function
on a compact Riemann surface M of genus
the analytic mapping
f: M --> ]P
exhibits M as an r-sheeted
branched covering of P with total branching order Proof.
Writing the divisor of the function r
form
¢(f) =
Z
i=1
plex line bundle
(1-pi- 1-q ) 1
where
9 = 5
=
y..5
pl
fC,fl E r(M, CL(g))
Pii
J q j3
2(g + r
f
such that
in the
consider the com-
ql...5qr , where
5
pr
point bundles considered in §7(c).
g , then
are the p
There are analytic sections 1-pi
,.9-(fC)
and ;"(fl) _
r
E 1-qi ; and
The functions
f = f0/fl .
fC
and
fl
have no
i=1
common zeros, and the mapping
morphic function
f
defined by the mero-
f: M --- ]P
coincides with the mapping
constructed in Theorem 24 .
(f0,f1): M T ]P
The desired result thus follows
immediately from that theorem. Corollary 2.
If
9 e H'(M, (Q *)
is a complex line bundle
on a compact Riemann surface M , such that
c(g) = r
and
7(e) > 2,
then to any pair of linearly independent analytic sections f0,fl e r(M, (9(e))
there is canonically associated a complex
analytic mapping
f = (fO,f1): M -. 1P which exhibits M as a branched covering of sheets.
]P
of at most
r
s
Let - =
Proof.
E
i=1 f0 and
zeros of the functions
choose an analytic section
be the divisor of the common fl , counting multiplicities; and
g e r(M,c1(9 (Ti))
such that J-(g) _
'r0 .
of the line bundle
Then
f0/g
and
fl/g
n =
are complex analytic sections of the complex line bundle
9-1 , and T1
these sections have no common zeros; so by Theorem 24+ the mapping
exhibits M as a branched covering of
(f0/g, fl/g): M -? P having
c(g 1) = r-s < r
of the divisor A-0 (f0/g, fl/g)
and
sheets.
Note that outside of the points
the function (f0,fl)
g
is non-vanishing, so that
define the same mapping of M to
the mapping is thereby canonically determined by the sections and
fl
]P
IF
f0
alone.
Corollary 3.
Any compact Riemann surface M can be
represented as a branched covering
f: M ---> IP
of the projective
line; the genus
g
of M , the number
branching order
b
are related by b = 2(g +r -1)
Proof.
r
of sheets, and the total .
Since every compact Riemann surface admits a non-
constant meromorphic function by the fundamental existence theorem,
Theorem 12, this assertion follows from Corollary 1; it is merely inserted for the sake of explicitness.
It is clear that any finitely-sheeted branched
Remarks.
covering of the projective line
]P
is a Riemann surface, with a
unique complex structure for which the covering mapping is an analytic mapping.
Then the genus can be calculated from the
branching order and the number of sheets by applying the formula in Theorem 24+.
The genus can also be calculated directly in a
purely topological manner, without reference to the analytic structure, as follows.
Letting
f: M T IP
covering, triangulate the surface images under
f
be an r-sheeted branched in such a manner that the
IP
of the branch points are vertices of the triangu-
The triangulation can then be lifted back to a triangula-
lation.
tion of M under the mapping the triangulation of P
f ;
it is only necessary to assume
fine enough that the interiors of the
one- and two-simplices are homeomorphic to each component of their f
letting ni be the number of i-simplices
in the triangulation of
IP, it is evident that the induced triangu-
lation of M will have
rnC-b
inverse images under
and
two-simplices.
rn2
zero-simplices,
rn1
one-simplices,
Thus the Euler characteristics (see
Seifert-Threlfull, Lehrbuch der Topologie, §23, (Chelsea, 1947))
of
and M are related as follows:
IP
X(M) = (rno-b) - (rnl) + (rn2) = r(no-n1+n2) -b = r-X(IP) - b . On the other hand, these Euler characteristics are also given by
X(M) = 2- 29 ,
X(]P) = 2 , where
g
is the genus of M -
Hence b = 2(r + g -1) , the desired
formula.
If M
is a compact Riemann surface of genus
g , it
follows from the Riemann-Roch theorem (recalling in particular
the table in formula 14 of §7) that g
for which
y(g) > 2
for any line bundle
c(g) = g + 1 ; hence by Corollary 2 of Theorem 24
the surface M can be represented as a branched covering of of at most
g+ 1
sheets.
IP
This is far from being the best pos-
sible result in general; we shall return to this question again
later, but for the present merely consider some simple results relating to the Weierstrass points on the surface.
For any point
be the least non-gap in the Weierstrass gap
p e M , let
r
sequence at
p ; it then follows from Theorem 14+ that
where
tp
is the point bundle associated to the point
M can be represented as a branched covering of F r
sheets.
p
f
having as its only singularity a pole at
of at most
p
on the surface
of order precisely r ,
considered as an analytic mapping
f: M - ]P
will exhibit M
as an r-sheeted branched covering of F ; the point
p e M will
be the only point of M covering the point at infinity on and hence will be a branch point of order Conversely, it is clear that whenever
branched covering such that a point
at
r-1 , then r
p ; for the image
at infinity on F sition of
M ,
From Corollary 1 of Theorem 24+, the function
by Theorem 14+ again.
order
Thus
.
Actually, an even more precise assertion can be made.
There will exist a meromorphic function
f
y(ip) = 2
r-l
IP
on the surface.
f: M ---> JP
is an r-sheeted
p e M is a branch point of
is a non-gap in the Weierstrass gap sequence f(p) a ]P
can always be taken to the point
by a projective transformation, and the compo-
f with that projective transformation will be a mero-
morphic function whose sole singularity is a pole at p of order r.
For a general point p e M the first non-gap value is r = g+l ; however when
g > 1
there are always Weierstrass points,
and at any such point the first non-gap will satisfy the inequality 2 < r < g .
At a normal Weierstrass point the value will be
and at a hyperelliptic Weierstrass point the value will be
r = g,
r = 2
.
In the latter case considerably more can be asserted, as follows. Theorem 25.
A compact Riemann surface of genus
g > 1
is hyperelliptic if either of the following two conditions holds: (i)
(ii)
the surface has a hyperelliptic Weierstrass point; the surface has a complex line bundle and
y(9) = 2
I
with
c(g) = 2
.
The hyperelliptic surfaces are precisely those which can be represented as two-sheeted branched covering surfaces of the projective
line P , and which are of genus
g > 1 ; the branch points are
precisely the Weierstrass points, all are hyperelliptic Weierstrass points, and there are
2(g+l)
of these points.
Note first that condition (i) implies condition
Proof.
(ii); for if p
is a hyperelliptic Weierstrass point on the
Riemann surface M , then
c(e2) = 2
and
y(t2) = 2
.
Now if
condition (ii) holds,' it follows from Corollary 2 of Theorem 24
that the surface M can be represented as a branched covering of ]P
with at most
2
sheets; and since genus
M # P , there will be exactly
2
sheets.
will necessarily have branching order the total branching order is gether
value
2(g+l)
r = 2
g > 1
means that
Each branch point
1 ; and since by Theorem 24+
b = 2(g+l) , there will be alto-
of these branch points.
At each branch point the
will be a non-gap, as noted in the above discussion;
hence all these branch points will be hyperelliptic Weierstrass points.
Recalling Theorem 16, it follows immediately that these
are all the Weierstrass branch points, and hence the surface is a hyperelliptic surface.
Since this argument only used the fact
that M could be represented as a two-sheeted covering of P
all the assertions of the theorem have been proved.
If M is a compact Riemann surface of genus then for any line bundle
t
Riemann-Roch theorem that
with cQ) = 2
it follows from the
7(e) = 2 ; so that M
sented as a branched two-sheeted covering of
sarily 2(g+l) = 4 branch points.
g = 1 ,
IP
can be repre, having neces-
This again illustrates the
similarities between elliptic and hyperelliptic Riemann surfaces.
If M has genus
g = 2 , then necessarily M
To see this, recall that for genus
g = 2
is hyperelliptic.
the Weierstrass gap
sequence has the form 1 = p1 < p2 < 4 , so that either
(and p
is not a Weierstrass point) or
p2 = 3
(and p
p2 = 2
is a
hyperelliptic Weierstrass point); since M has at least one Weierstrass point by Theorem 15, it follows immediately that M is hyperelliptic. genus
g > 2
genus
g
g
We shall see later that not all surfaces of
are hyperelliptic; and also that the surfaces of
can be represented by branched coverings of fewer than
sheets, if it is not required that all the sheets meet at some
point.
(b)
The preceding representation of a compact Riemann surface
as a branched covering of the projective line can be used to provide a useful description of the global meromorphic functions on the surface.
The set m
m
of all meromorphic functions on any
Riemann surface M is a field, under the operations of pointwise addition and multiplication of functions.
The field "I M
con-
tains the subfield C of complex constants; and for any element
f e
the field
M
also contains the subfield
M
rational functions of f . and for any element
coefficients in E
C(f)
(Recall that for any fields
E C F
x e F , the set of all polynomials in
and x
is the
x with coefficients in E,
set of all quotients of polynomials in
or in other words the set of all rational functions of x , .
x with
is an integral domain denoted by E[x] ; and
the smallest subfield of F containing both E
denoted by E(x)
of
and is
For the elementary properties of fields which
will be used here, see for instance B. L. van der Waerden, Modern
Algebra vol. I, (Frederick Ungar Co.)
N.Y.,
191+9).)
As a simple
preliminary, note the following result. Lemma 30.
If
on the projective line Proof.
Let
projective line P . mapping points
f: P --> P p # q
in
I'
is merbmorphic on P case that either p
f
is a meromorphic function of degree 1
7P
, then MIP =
C(f)
.
be the inhomogeneous coordinate on the
z
Note that by Corollary 1 to Theorem 24, the is an analytic homeomorphism.
, the function
(f(z)-f(p))/(f(z)- f(q))
and has divisor precisely or
q
Then for any
l-q ; in
is the pole of f , the obvious modi-
fications of this formula will be left to the reader.
If g e NP
is any meromorphic function, with divisor $(g) = Ei(pi- qi) , it is clear that
g (z) = C ]I for some constant
C , hence that
f(z) - f(pi)
i fz - fqi) g e C(f)
to prove the assertion of the lemma.
.
This suffices
It should be noted that the inhomogeneous coordinate
on P
z
can be considered as a meromorphic function of degree
on P ; and hence by the above Lemma,
P
= C(z)
J.
The follow-
.
ing generalization of this lemma is quite straightforward. Theorem 26.
Let M be a compact Riemann surface, and
f e ChM be a meromorphic function of degree Then for any function P(x,y) a C[x,y]
g e
on M .
r > 0
there is a polynomial
M
in two variables, of degree at most
second variable y , such that
P(f,g) = 0
.
in the
r
(Note that
P(f,g)
is a well-defined meromorphic function on the Riemann surface M .) Proof.
By Corollary 1 to Theorem 24+, the function
considered as a mapping
f: M ---> P
exhibits M
branched covering of the projective line P .
f
as an r-sheeted
The image under f
of the branch points then forms a finite set of points g1,...)gs e r , and the mapping
f: M - Ui f I(qi) --> ]P - Ui qi is an r-sheeted covering space in the ordinary sense. point
q e P
which is not one of the branch points
For each q1'
...,qs
select a contractible open coordinate neighborhood V of
P which is regularly covered under the mapping r f-'(V) = U Ui , where the i=1
homeomorphic to V ; morphisms such that
let
(pi: V ---> Ui
piof: Ui T Ui
FV(z,Y) =
Thus
be the analytic homeo-
is the identity for each i.
on M , introduce the function
g
r
(1)
.
in
are disjoint open subsets of M
Ui
For any meromorphic function
f
q
i 1 (Y-goTi(z)) ,
this function is a polynomial in y of degree cients are meromorphic in
course, when p e f l(V)
for
z
z e V .
r , and its coeffi-
By construction of
,
FV(f(p),g(p)) = ii (g(p) - go(Pi.f(p)) = 0 , since
if p e Ui -
goCp. f(p) = g(p)
The same construction can
be carried out in any other such coordinate neighborhood W , FW(z,y)
yielding another function
of a similar form.
intersection v 11 W the mappings
Ti,V
and
cPj,W
In an
coincide in
some order; the coefficients of the polynomial (1) are the elementary symmetric functions of the values independent of the ordering.
Therefore
goc)i(z) , and hence are
FV(z,y) = FW(z,y)
z e v n w ; and hence there is a well defined function
a polynomial in y of degree
g
F(f,g) a 0
.
If the
is analytic at the points f'() , it is clear that
the coefficients of the polynomial functions of
F(z,y)
with coefficients which are mero-
]P- Ui qi , such that
morphic functions on function
r
for
z
F(z,y)
are bounded analytic
in a punctured disc centered at
qi ; hence by
Riemann' s removable singularity theorem, the coefficients remain
analytic at the point
qi
.
It is a straightforward matter,
which will be left to the reader to verify, that the coefficients
of F(z,y)
are meromorphic at those points
f1() contains poles of meromorphic on all of inhomogeneous coordinate F(z,y)
g .
properties.
such that
Thus these coefficients are
, hence are rational functions of the z
of F, by Lemma 30; multiplying
by a suitable polynomial in
polynomial P(z,y)
qi
z
will therefore yield a
in two variables, with all the desired
Corollary.
The field of meromorphic functions on a
compact Riemann surface is an algebraic function field in one variable over the complex numbers, that is, is a finite algebraic
extension of a simple transcendental extension of the field C . Proof.
f e % is any non-constant meromorphic
If
M , the field
function on the Riemann surface
C(f)
is a simple
transcendental extension of the field C ; for otherwise
f would
be the root of a polynomial with coefficients in C , hence would necessarily be a constant.
If
select any meromorphic function
C(f)
of degree at most
meromorphic function
select another function
g2 a
E2 = C(f,gl,g2) ; since
g2
at most
r , the field
degree at most tinued.
r
E2
as well.
E1 ' IM -
and consider the
is an algebraic extension
El
r , where If
f .
M - C(f)
g1 e
field E1 = C(f,g1) ; by Theorem 26, of
is not the full field N M
C(f)
r
is the degree of the
is not the full field 'h7 M
E1
and consider the field
is algebraic over
C(f)
of degree
is an algebraic extension of E1
This process can of course be con-
However, by the theorem of the primitive element (cf. van
der Waerden, page 126), the extension Em = C(f,gl,...,gm) C(f)
of
can be generated by a single element
for some complex constants have degree at most
r
over
of
c1g1 + c2g2 +...+ cmgm
cl,...,cm ; so that actually E. must C(f)
for all m .
The process then
necessarily stops after finitely many stages, and the result is thereby demonstrated.
Now on a compact Riemann surface M select any two meromorphic functions
f,g which generate the function field of the
surface, that is, which are such that 1
M
= C(f,g) ; and let
P(x,y)
be the polynomial such that
P(x,y)
can always be taken to be an irreducible polynomial.
P(f,g) m 0 , noting that It
is clear that this polynomial completely describes the function field of the Riemann surface, as an abstract field.
It is indeed
even true that the polynomial describes the Riemann surface itself, in a sense.
It is more convenient for this purpose to pass from
the polynomial P(x,y) polynomial. if
to a naturally associated homogeneous
Formally, write
x = t1/t0
and y = t2/t0 .
Then
n is the degree of the polynomial P(x,y) , consider the homo-
geneous polynomial of degree
in three variables defined by
n
P0(t0,tl,t2) =
0
(tl/t0,t2/t0)
This will be called the homogeneous form of the polynomial
it is canonically determined by
P(x,y) , and the original poly-
nomial can be recovered by noting that sidering
(tO,t1,t2)
P(x,y);
P(x,y) = PO(1,x,y)
.
Con-
as homogeneous coordinates in two-dimensional
complex projective space
IP 2 , although P0(t0) tl,t2)
well-defined function on
]P2 , its zero locus is a well-defined
is not a
subset of P 2 ; for if P0(tO,tl,t2) = 0 , then from homogeneity it follows that
PO(tt0,ttl,tt2) = t'P0(t0)tl,t2) = 0
.
The sub-
set
loc PO = ((to) tl) t2) e P 2 PO(t0,tl,t2) = 0) is called an algebraic plane curve of degree
polynomial
PO .
n , defined by the
(It is assumed that the reader is acquainted
with the elementary properties of projective spaces; see for
instance W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry,
volume I, chapter V, (Cambridge University Press, 1953), for a treatment of this topic.) in
]P2
In the coordinate neighborhood
, with local coordinate
curve is given by the equation algebraic plane curve
loc P
0
x = tl/t0
and y = t2/t0 , the
P0(l,x,y) = P(x,y) = 0 ; so the
in projective space is in the
natural sense just the extension of the curve
Note that the intersection of the curve be P in
g,2
defined
P(x,y) = 0
in the ordinary space ( of the two complex variables
infinity t0 = 0
to # 0
x,y
.
with the line at
0
consists of a finite number of points;
these are the points with homogeneous coordinates satisfying P0(O,t1,t2) = 0 , or in terms of the inhomogeneous coordinate t = t2/t1
on the line at infinity, the points
PO(0,1,t) = 0
Removing these finitely many points from loc PO P(x,y) = 0 set
loc PO
yields the curve
In the relative topology as a subset of
.
1P2 , the
is just the compactification of the curve
P(x,y)= 0 ,
obtained by adding a finite number of points to that curve. Lemma 31.
To any algebraic plane curve there is canonically
associated a compact Riemann surface. Proof.
Consider first that portion of the curve
in the coordinate neighborhood set
to # 0 ; this is just the point
loc P = ((x,y) a C2IP(x,y) = 0)
nomial in y with coefficients in is irreducible, and let D(x)
is a polynomial in
D(x)
loc PO
.
View
P(x,y)
as a poly-
C(x) , assuming first that it
be its discriminant; note that
x , and let A = (xi,x2,...)
finite number of zeros of that polynomial
D(x)
.
be the
For each value
x0 / A the polynomial P(x,y) r
r
distinct roots, where
is the degree of that polynomial in the variable y ; and letting
these values be
yl,...,yr , since they are simple roots it follows
7P(x0,yi)/ay J 0
that are
will have
r
.
By the implicit function theorem, there
complex analytic functions
in an open neighborhood of x0
cPi(x0) - yi The points
(x,(p i(x))
cpl(x),
i = 1,.. .,r , defined
and such that
and
P(x,(pi(x)) m 0 .
are thus all the points for which P(x,y)= 0.
It is then clear that under the natural projection
v: C 2
e
defined by ir(x,y) = x , the curve ((x,y) a eIP(x,y) =0, x J A ) is an r-sheeted covering space of the set C - 0 = (x a CID(x) / 0). This provides that portion of the curve with a Riemann surface structure.
Then for a point
x1 e 0 , the curve
P(x,y) = 0 will
locally provide an r-sheeted covering of a punctured disc centered at the point
xl .
These coverings are all well known, however;
each connected component can be completed as a branched covering of the full disc centered at
xl , and the Riemann surface struc-
ture extends uniquely to this completion.
This associates"a
Riemann surface to that portion loc P C loc PO the given coordinate neighborhood.
of the curve in
Note that, except possibly at
these branch points, the underlying point set of the associated Riemann surface can be identified with the point set the identification is a topological homeomorphism.
loc P ; and
Moreover,
again excepting the finitely many branch points, it is evident
from the local parametrizations y = si(x) analytic functions on
loc P
that the germs of
in terms of its structure as a
Riemann surface are precisely the restrictions to be P
of germs
of analytic functions of two complex variables defined in a neighborhood in the projective plane
1P2
.
Recalling from lemma 2
that a Riemann surface structure is determined completely by the sheaf of germs of holomorphic functions on the surface, and observing from Riemann's removable singularities theorem that a mapping between two Riemann surfaces which preserves the sheaves of analytic functions except for a discrete point set is neces-
sarily an analytic equivalence, it follows that the Riemann surface structure is uniquely determined, independently of the choices made in the above construction.
Now a similar construction
can be carried out in the other coordinate neighborhoods of
IP2
;
by the above uniqueness observation, these complex structures necessarily match in the intersections of the coordinate neighborhoods, and hence define a unique compact Riemann surface associated loc PO .
to the algebraic curve
In case the polynomial
P(x,y)
is reducible, each irreducible factor separately determines a compact Riemann surface.
To illustrate the manner in which the Riemann surface associated to
loc P
differs from the point set be PO , con0
sider the trivial case of the polynomial
P(x,y) = xy .
The
Riemann surface corresponding to that curve consists of two disjoint copies of the complex line
C ; they are imbedded in
in such a way that they intersect, but that is not reflected in the Riemann surface structure.
The analytic functions on the
Riemann surface are not the restrictions to the curve
loc P
of
analytic functions of two complex variables at the origin; for the restrictions would necessarily have the same value at the intersection of the two lines, while the functions on the Riemann surface do not.
For the case of an algebraic curve with singularities,
such as the curve defined by the equation P(x,y) = y2 - x3 which has a singularity at the origin, the Riemann surface and the curve agree as point sets; but still the germs of analytic functions on the Riemann surface do not consist of the restrictions to the curve of germs of analytic functions of two variables at the origin. This leads further into the properties of singularities of algebraic curves than time permits,for adequate treatment; the reader is referred to the standard literature on algebraic curves for a more extensive discussion. Theorem 27.
Let M be a compact Riemann surface, and
f,g a ?1l M be meromorphic functions generating the function field
of M ; and let P(f,g)
P(x,y)
be the irreducible polynomial such that
There is then a canonical analytic homeomorphism
0 .
from the Riemann surface
M onto the Riemann surface associated
to the algebraic plane curve defined by the polynomial P(x,y) Proof.
Let
g e
H1(M, 6*) be a complex line bundle on
the Riemann surface M with a non-trivial holomorphic crosssection
f2 = fog i = 0,1,2
f0 a r(M, Q (g))
such that the functions
are holomorphic; then of course .
The line bundle
that the three functions
g
it suffices to select the section
and
fi a r(M, 1 (¢)) ,
and section
f0,fl,f2
f1 = f0f
f0
can be so chosen
have no common zeros on M f0
such that its divisor is
the least for which PO(tO,tl,t2)
note that
f0f
and
fOg
are holomorphic.
Letting
be the homogeneous form of the polynomial P(x,y)
PO(f0,fl,f2) = f (fl/f0, f2/f0)
Thus the mapping from M into
1P2
f-'P(f,g)
0
defined by g,2
p e M -. (f O(p),fl(p),f2(p)) a takes the Riemann surface M to the point set
loc P
0
the algebraic plane curve defined by the polynomial
of
C ]P 2
P .
It is
readily verified that this defines a complex analytic mapping from
M to all points in the Riemann surface associated to the curve, except the branch points; and applying Riemann's removable singu-
larities theorem yields an extension to all of M . will be left to the reader.
The details
Thus there is defined a local branched
covering from M to the Riemann surface associated to the curve P(x,y) ; and since both are compact, it follows easily that this is an r-sheeted branched covering for some index
r
.
The mero-
morphic functions on M separate points, in the sense that if
M
p,q e M and p
q
such that
h(q) ; for there is always a meromorphic function
h(p)
there is a meromorphic function h e
with a pole only at the point p
.
Since
f,g
generate the mero-
morphic functions, then these two functions generally separate
points as well; thus the index r = 1 , and the mapping from M to the Riemann surface associated to the curve
P(x,y)
is one-to-
one, thereby completing the proof. Corollary.
Two Riemann surfaces are analytically equiva-
lent if and only if their fields of meromorphic functions are isomorphic as abstract fields.
Proof.
If M,M'
are two Riemann surfaces with isomorphic
function fields, generators of these fields can be chosen such that they satisfy the same irreducible polynomial equation; the desired result follows immediately from Theorem 27 then. The preceding Corollary shows that the investigation of compact Riemann surfaces can be reduced to the investigation of algebraic function fields in one variable over the complex numbers, or of algebraic plane curves.
The equivalence concept for func-
tion fields is just isomorphism as abstract fields, but is slightly more complicated for algebraic curves. Pt(x',y')
If
P(x,y)
and
define algebraic curves, then these should be considered
as equivalent when their underlying function fields are the same.
This means that
xt,yt
(considered as meromorphic functions on
the curve) must be rational functions of
x,y , and conversely;
this equivalence concept is known as birational equivalence.
The
algebro-geometric form of the study of compact Riemann surfaces can be phrased as the problem of studying birationally invariant
properties of algebraic plane curves.
This was the original form
in which the subject was studied, and the reader is referred to the standard works on algebraic geometry for further reading. The principal interest in these lectures has been the analytic aspects of the subject, so the algebraic line will be pursued no
further.
Let M be a compact Riemann surface of genus
(c)
and let h1,h2...,hg a r(M, a(K)) Abelian differentials. sections
hi
g > 1 ,
be a basis for the space of
It was noted earlier (page 119) that these
have no common zeros on the surface.
Thus if
za
is a local coordinate flapping in an open set U. C M , and if
hi(z) are the analytic representations of the sections
hi
in
that coordinate neighborhood, then the values
can be viewed as the homogeneous coordi-
(hla(za),...,hga(za))
nates of points in the projective space
g-1
IP
this leads to a complex analytic mapping
of dimension g-1;
Ha: Ua -> 'P g-1 .
these mappings are related by Ha(za) _
Note that in Ua fl Up
(hla(Za),...,hga(Za)) _ (Ko,(z,,)h1,(z
H,(z.) ; this therefore yields a global analytic mapping
The mapping H
H: M ---> IP g-1 .
and the image
g-1
H(M) C IP
is called the principal curve associ-
ated to the Riemann surface M . basis for
T(M,
'.(K))
is called the principal mapping,
Note that choosing a different
has the effect of replacing the principal
mapping H by a non-singular linear transform of H , or equivalently, of following the mapping H by a non-singular projective transformation in
IP
g-1 .
Thus the principal mapping and the
principal curve are determined uniquely up to a non-singular projective transformation in Theorem 28 (a). genus
g > 1
mapping
g-1 IP
If M is a compact Riemann surface of
and M is not hyperelliptic, then the principal
H: M'--> IP
g-1
is a one-torone non-singular complex
analytic mapping, and the principal curve sional complex analytic submanifold of
H(M)
IP g
-1
is a one-dimen-
Proof.
That the mapping
H: M T 7P 9-1
is a complex
If H were not
analytic mapping is obvious from the definitions. one-to-one, there would be two distinct points
p,q e M such that
H(p) = H(q) ; and by a non-singular projective transformation in g-l
that image can be taken to be the point
IP
(1,0,...,0) 6 ]P Let
g-1 .
J = CPtq , and let
Thus
H(p) = H(q) _
hi(p) = hi(q) = 0
f e T(M, -(g))
for
be the standard non-
trivial section, with divisor 9(f) = for
hi/f
i = 2,...,g.
The functions
i = 2,...,g , then linearly independent complex Kg-1
analytic sections of the complex line bundle y(Kg-1) > g-1
.
, so that
By the Riemann-Roch Theorem,
y(e) = y(Kg-1) + c(g) + 1-g > 2 ; but since
e(g) = 2 , it would
.follow from Theorem 25 that M is hyperelliptic, a contradiction.
Therefore the mapping H
is one-to-one.
The Condition that the
mapping H be non-singular is just that at each point of M at least one of the coordinate functions of the mapping H be non-
If p e Ua
singular, that is, have a non-vanishing derivative. and
za
is a local coordinate in
in homogeneous coordinates by if say
Ua , the mapping H
za ---;- (hla(za),...,hga(za)) ; and
h1a(p) # 0 , then in terms of the standard inhomogeneous
coordinates around H(p) a ]P g-1 , the mapping H
by
is given
(h2a(za)/hla(za),...,h9a(za)/hia(za))
za T
H were singular at
is described If the mapping
p , then necessarily
h1a(P)hia(P)- hia(P)hia(P)
= 0
for
i = 2,...,g
(P)2
hla Since
h1a(P) / 0 , then writing
ha(za) = (hia(za))
and
hh''(za) _ (hia(za)) (
(p))W(p))
follows that
as on page 117, it follows that the matrix
has rank
p = 1 .
Then from Lemma 17 it further
y(CP) = 2 ; but since c( 2) = 2
j.
that M is again hyperelliptic, a contradiction.
Theorem 25 implies
The mapping H
is therefore non-singular, and as an immediate consequence of that, the image curve
H(M) C IP g-1
is a one-dimensional complex ana-
lytic submanifold of the projective space, concluding the proof. A few simple properties of the principal curve of a nonhyperelliptic Riemann surface are as follows.
First, the prinPg-1
cipal curve does not lie in any proper linear subvariety of For letting
g-l
(tl,...,tg)
if the principal curve
be homogeneous coordinates in H(M)
1P
for all
p e M ; but this is impossible if not all the constants
hi a r(M, .(K))
vahish, since the sections
H(M)
,
were contained in the linear sub-
variety Eiaiti = 0 , then necessarily Eiaihi(p) = 0
pendent.
.
ai
are linearly inde-
This means that a linear subvariety meets the curve
in a finite number of points only; indeed, the intersection
consists of precisely 2g - 2 letting
cp(t) = Eiaiti
plex line bundle on
,
points, counting multiplicity.
For
as a complex analytic section of a com-
g-l IP
, the restriction of
to
H(M)
has a well-defined total number of zeros, which is the desired intersection multiplicity; but that restriction is the element
cp(H(p)) = Eiaihi(p) a r(M, 0 (K)) , and since c(K) = 2g -2 , follows from Theorem 11 that counting multiplicities.
cp(H(p) ) has altogether 2g - 2
zeros,
It is known that any complex analytic
Pg-l
subvariety of
it
is an algebraic variety; this is Chow's
theorem, (and a proof can be found for instance in R. C. Gunning and H. Rossi, Analytic Functions of Several complex variables (Prentice-Hail, 1965), page 170).
The principal curve is there-
fore a non-singular algebraic curve in the projective space IPg-l; since the degree of a curve is its intersection multiplicity with a general linear hyperplane, it follows from the above remarks that
is of degree
H(M)
2g -2
.
In the case of a hyperelliptic Riemenn surface M , the situation is quite different.
f: M .- IP
To any mapping
exhibits M as a two-sheeted branched covering of
which
as in
IP
Theorem 25 there is canonically associated an analytic automorphism
0: M -> M of period two.
Omitting the branch points,
is an ordinary two-sheeted covering mapping, and operation of interchanging the sheets.
f: M ---> 3P
is just the
a
The operation extends
analytically to the branch points in an obvious manner; for a local coordinate
branch point f(z) = z?
,
can be chosen in an open neighborhood of the
z
in M such that the mapping
z = 0
and in that neighborhood
ez = -z .
The fixed points of
2g+ 2
Weierstrass points on the surface
Such a mapping e
0
M ---> M/
f: M -. P ; for
M , and the mapping
(1,B)
f
92
is
are precisely the
M , since by Theorem 25
these are precisely the branch points of the mapping The mapping
has the form
As noted,
will be called a hyperelliptic automorphism. the identity mapping.
f
f: M --;h IP.
can be described as the quotient mapping is a group of analytic automorphisms on
is equivalent to collapsing M into the
quotient space under this group of automorphisms.
It is thus
obvious that a compact Riemann surface
M
is hyperelliptic pre-
cisely when it admits a complex analytic automorphism of period two such that the quotient space the fixed points of
e
e: M --k M
ml(e) = IP; and although
are unique, it is not yet evident that there
is a unique such mapping Theorem 28 (b).
e
on any hyperelliptic Riemann surface.
If M is a hyperelliptic Riemann sur-
face, the principal mapping
H: M T IP
g-1
can be factored as
the composition of the quotient mapping M T M/(e) modulo a hyperelliptic automorphism, and the canonical non-singular complex
analytic mapping IP -- ]P9_1 defined by (zo, z1) --> (zo-1,
(2)
zo-2z1, ..., z0zI-2,
z1-1)
in terms of the standard homogeneous coordinates. Proof.
Let
f: M
-->
IF
be a two-sheeted branched
W on
covering such that the point with inhomogeneous coordinate
P
Then considering
is not the image of a branch point.
as
f
a meromorphic function on M , it follows that for some points
j (f) = 1-r1 + 1-r2 -
ri, qi e M
Considering df e P(M, !] 110) = T(M, 777 (K)) as a meromorphic differential form on M , the zeros of branch points of the mapping
occur precisely at the
df
f , and
df has double poles at the
points qi a M ; therefore where
pi e M are the branch points of
points on M .
f , hence the Weierstrass
Select a meromorphic function
A (g) = 1-pl +'...+
(g+l).q1
- (g+1)-g2
existence of a function of this sort, let
on M
g .
such that
(To show the
qi = f(pi) a IP
,
and
consider the two-valued meromorphic function y on F defined by y = (z-a1) ... (z-a2g+2) dinate
on F.
z
in terms of the inhomogeneous coor-
The function y
points at the points
ai ; hence
y
is two-valued and has branch can be viewed as a single-
valued meromorphic function on the two-sheeted covering surface M, and as such will be the desired function.) that for
i = 1,...,g
It now follows readily (pi = f1'-ldf/g
the differential forms
are
holomorphic on the Riemann surface M ; for J-((p1) = (i-1)
,
(f) + A (df) - 4(g) _ (i-1)-rl + (i-1).r2 +
(g-l).q2
+
Since these differential forms are evi-
.
1,0).
dently linearly independent, they determine a basis for r(M, 0.
In terms of this basis the principal mapping clearly has the form
H(p) _ (1'f(p)'f2(p),...,f -1(p))
(3)
This shows first of all that the mapping H through the mapping
f ;
can be factored
and rewriting (3) in homogeneous form to
avoid considering meromorphic functions, it is evident that
the composition of
f
and the mapping (2).
H
is
It is a straight-
forward' matter, verification of which will be left to the reader,
to see that (2) is a one-to-one non-singular complex analytic
mapping of
g-1 ]P
onto
Corollary 1.
IP
and the proof is thereby concluded.
On a hyperelliptic Riemann surface M the
hyperelliptic automorphism Proof.
,
e: M
M is unique.
The principal mapping is unique, up to composition g-1
by a non-singular projective transformation in e
IP
; and since
is determined by the principal mapping, as in Theorem 28 (b),
the Corollary follows immediately.
Let M be a hyperelliptic Riemann surface
Corollary 2.
of genus
g , with the hyperelliptic automorphism
and let
p1,...,pg-1 be any points on the surface
e: M - M ; M .
Then the
canonical bundle of M is given by K = tpltepl...tpg-ltepg-1
Proof.
f: M
Let
-->
p,ep e M be the points which map into
branched covering, and let the point
oo a IP
ing; so that p
F be the standard two-sheeted
in the standard inhomogeneous coordinate coverand
ep
are the poles of
f ,
considering
The differential form df/g
a meromorphic function on M .
f
as
con-
structed in the proof of Theorem 28 (b) is holomorphic on M and and therefore
has the divisor A (df/g) = (g-1) -p + 'P-1te1
K =
.
This is the particular case of the Corollary in
which p1 = ... = pg-l = p
.
a = f(p) , be the images of these points in
g* on
meromorphic function
function f: M -.
7P
,
it is clear that
g
There is a
g* to a meromorphic
will have the divisor
epi)
(g) _
.
M, by means of the mapping
on the Riemann surface
g
IP
with divisor
IP
and lifting
9.(g') _
ai = f(pi) ,
In general, let
.
It then follows that
g-1 g-1
p
ep =
and hence the Corollary follows from the special case just proved.
A representation of Riemann surfaces which is closely associated to the principal curve is the following. 1P1,...,(P g
Let
a r(M, &-Z'0), be a basis for the space of Abelian dif-
ferential forms on the surface; and let but fixed base point on the surface. Hl'(M,Z) , the cohomology classes
by vectors
(woi) e G26
fl = (wji)
po e M be an arbitrary
Choosing a basis for are represented
Scpi a H1(M,©)
2g x g matrix
as on page 141+; and the
is the associated period matrix of the Abelian differ-
entials on M .
Recall that the Jacobi variety J(M)
of the
Riemann surface M is the compact complex torus N(M) = CgItSt Now for any point p e M , select any path
.
from p
to
0
p
g.
in
M ; and introduce the element
(4)
If T
T.l
is any other path from p 0 to p , then
is a closed loop from p
dual basis for
0
to
po
(T j )
e
Z2
g .
Thus
a E ,j=1
E =1
a
tfleg .
(fXWi)
in
,JA
T.l = X+ T
where
in M ; and in terms of the
HI(M,Z) , the homology class of
vector
sented by 2g
/tsdg = N( M)
O(p) = (IX(Pl,..., j,Ncpg) a
(M)
T
will be repre-
(f %jcp) _ ( f )'cpi + fT(Pi)
, since
The mapping 0: M T N(M)
is thus well-
defined, being independent of the choice of the path
X.
.
This
mapping is called a Jacobian mapping of the Riemann surface M Note that the mapping is independent of the choice of bases for
P(M, &"0) and H1(M,C) , in the obvious sense; but depend on the choice of the base point p
0
0
does
e M , a change in the
base point corresponding to a translation in the Abelian group ti(M).
Theorem 29.
If M is a compact Riemann surface of genus
g > 0 , then the Jacobian mapping
0: M --- > J(M)
is a one-to-one
nonsingular complex analytic mapping, and the image is a one-dimensional complex analytic submanifold of That the mapping
Proof.
0
D(M) C J(M)
N
J(M)
is a complex analytic mapping
is obvious, since the integrals in (4+) are complex analytic functions of the limits of integration, at least locally.
mapping
were not one-to-one, there would exist distinct points
0
gl,g2 e M
If the
such that
IP(gl) = !D(q2) ; and in terms of the explicit
form (4+) for that mapping, it would follow that for an are
from
q1
to
T
q2
(1T(Plf ..., Lr(Pg) = 0 E Cg/ts?.eg = J(M) . Then from Abets Theorem (Theorem 18) it would further follow that
Cl = 1 , a contradiction (recalling the discussion on page 115).
p q
If
za
is a local coordinate mapping in a coordinate neighborhood
Ua C M , then writing
mapping
0
cpi = hia(za)dza , the condition that the
be singular at
za
is clearly just that
h1a(za) _ ... = hga(za) = 0 ; but this can never happen, as noted on page 119.
fore the image
Therefore the mapping O(M)
0
is nonsingular, and there-
is a one-dimensional complex analytic sub-
manifold of J(M) , completing the proof. Corollary.
If M is a compact Riemann surface of genus
g = 1 , then the Jacobian mapping isomorphism of Riemann surfaces.
0: M T J(M)
is an analytic
Proof.
Since
dim J(M) = g = 1
in this case, the Corol-
lary is an immediate consequence of the preceding Theorem.
This
provides a useful standard form for compact Riemann surfaces of
genus 1
.
The next stage of the discussion of Riemann surfaces would involve a more detailed investigation of these last mappings (the principal and the Jacobian mappings), leading towards Torelli's Theorem and the problem of moduli of Riemann surfaces.
Time has
run out, however, and this must be postponed to another time.
Appendix: the topology of surfaces.
It has been assumed that the topological properties of twodimensional manifolds, from the point of view of Cech cohomology especially, are familiar to the reader. be added here, in case that is not so.
A few words should perhaps Most books on Riemann sur-
faces begin with a discussion of the topology of surfaces, usually simplicial or singular homology theory with particular emphasis on the two-dimensional case; and the reader without this background can quite well consult one of these books.
(See for
example Lars V. Ahlfors and Leo Sario, Riemann Surfaces (Princeton University Press, 1960); George Springer, Introduction to Riemann
Surfaces, (Addison-Wesley, 1957); and of couse Herman Weyl, The Concept of a Riemann Surface, (English translation, AddisonWesley, 1964).)
The topology of surfaces, also from the point of
view of singular homology theory, is covered in H. Seifert and W. Threlfall, Lehrbuch der Topologie (Teubner, 1934; Chelsea, 1947); the fundamental group and covering spaces are also treated in detail there.
The 6ech cohomology groups of a compact surface (with coefficients in a constant sheaf, such as
Z or
C)
are isomorphic
to the singular or simplicial cohomology groups, and the cohomology groups can be viewed as dual to the homology groups; so the properties of the 6ech cohomology groups needed in these lectures can readily be derived from the discussion of the homology of surfaces in the books mentioned above.
(A more general dis-
cussion can also be found in Samuel Eilenberg and Norman Steenrod,
Foundations of Algebraic Topology, (Princeton Univ. Press, 1951).) More directly, the surface can be triangulated; and taking open neighborhoods of the closed two-simplices as an open covering of the surface, most of the results needed follow from a straightforward calculation.
Similarly, it follows quite easily that the
fundamental group as defined in §9 (b) is isomorphic to the fundamental group as more customarily defined, (as for instance in Seifert and Threlfall).
Referring to the discussion on page 186,
note that for an Abelian group
it follows that
G
Hom(7r1(M), G)/G = Hom(7r1(M), G) , and hence by Lemma 27, H1(M,G) = Hom(7r1(M),G) .
Indeed, since
Hom(7r1(M),G) = Hom(H1(M),G) , where
H1(M)
is the group
H1(M)
made Abelian, that is, is the quotient of tator subgroup; but
is Abelian,
G
7r1(M)
7r1(M)
by its commu-
is the first homology group of the
surface, so this is just the familiar duality between homology and cohomology.
In particular,
H'(M,C) = Hom(Hl(M),C) = Hom(7r1(M),C).
With this observation, the isomorphism
S:
r(M,(9Vl'°) ---> Hl(M,C)
discussed in §8 (a) can be put into a more traditional form as follows.
Let
(UO,U1,...,Un,UO)
terms of an open covering a closed path in the set
of the surface M ; and let U0 U U1 U...U Un from the point
representing the same element of morphism.
be a closed chain at p , in
Thus selecting points
7r1(M,p)
X be p ,
under the obvious iso-
pi a Ui , the are
taken as a union of smooth segments
Xi , where
Ui U Ui+1 from pi to pi+1 . If
cp a r(M,
Abelian differential form, select functions
Xi
X
can be
is an are in
&"0) is any fi E r(Ui, 0)
so
that hence
dfi= cp in Ui. fi+l
In Ui fl Ui+1 note that d(fi+1- f i ) = 0
- fi = ci,i+1
for some constant
suitably modifying the functions ality in supposing that i = 0,1,...,n-1 .
for
Ui fl Ui+1
Now the element of
Hom(7r1(M),C)
assigns to this loop
H1(M,C) ---> Hom(ir1(M),C)
,
r(M, CL" O) -4. H'(M,C)
and
is that which assigns to an Abelian
differential form p e r(M, 6-1'0) .
Therefore
r(M, 0'l 0) -> Hom(7r1(M),C) , derived from the
composition of the isomorphisms
f%cp
corre-
X the constant
fn - fo , which is evidently the value
the isomorphism
period
in
but this merely amounts to the constant
i=o ci' i+l ;
cn o =
fi = fi+l
8q' a H1(M,C)
n E
fi , there is no loss of gener-
The resulting function. is of course an indefi-
nite integral of p . sponding to
ci,i+l ; so upon
and a loop
% e 'rrl(M,p)
the
This justifies the period matrix terminology con-
sidered on page 145, and the discussion in the proof of Corollary 3 of Theorem 18.
Of course,
r(M, C9 1'0)
can be
replaced by the full space of closed differential forms on M , in the analogous discussion of deRham's Theorem in terms of the periods of differential forms.
Finally, a few words should be said about the intersection matrix on a surface, in connection with the discussion in §8 (c).
In the usual approach, the explicit form of the
intersection matrix
X
(recall page,l54) is derived in the proof
of Abel's theorem, or in the related discussion. and Sario, pages 319 ff., for instance.) essentially the following.
(See Ahlfors
The argument is
Suppose the surface M
is represented
by a polygon with pairs of edges identified, in the normal form, '(as in Seifert and Threlfall, pages 135 ff.):
bl
1
The elements
ai,bi
generate
Hl(M-)1; and dual generators can be
selected for the cohomology group
H (M,C )
.
Upon representing
these cohomology classes by closed differential forms e P(M, E 1) , this condition is that
(
Iaa j =si;
)
where
8i
i
i
is the Kronecker symbol.
intersection matrix
Xij = Ilai-a3 for
Ibaj=l aBj=o; lbaj=si
i
i
In terms of this basis, the
X has the entries
Xi, 3+g
i,J = 1,...,g , where
= Ilai-pj
g = genus of M.
Xi+
3+g =
IlaiJi
Upon applying Stokes
theorem a few times, it follows readily from the equations (*) that
X has the desired form.
The details can be left to the
reader, (compare Ahifors and Sario, pages 319 ff.).
Abelian differentials, 72 Abelian varieties, 151 Abel's theorem, 160 Algebraic plane curves, 234 Birational equivalence, 240 Branched coverings, 220 Canonical bundle, 78 Cauchy-Riemann equations, 40, 87 Chern class (characteristic class), 98 Cochains, 27 Cocyclea, 27 Cohomology groups, 28, 30 Cohomology sets (non-Abelian coefficients), 175 Connections (affine & projective, 169 Coordinate cohomology class of a structure, 176 Coordinate coverings, 1 -, complex analytic, 3 Coordinate transition functions, 1 Cross sections of line bundles, 53 Cup product, 147 Degree DeRham DeRham Direct
of a meromorphic function, 224 sequence, 68 theorem, 69 limit, 30
Distributions, 80, 90 Divisor, 48 - of meromorphic functions, 50 - of sections of line bundles, 56 Divisor class group, 53 Dolbeault sequence, 72 Dolbeault theorem, 72 Dolbeault-Serre sequence, 74
Eichler cohomology group, 207 Elliptic Riemann surface, 127 Exact cohomology sequence, 32 Exact sequence of sheaves, 25 Fundamental group, 186, 189, 252 Genus, 109 Geometric realization of a structure, 217 llyperelliptic automorphism, 244 Hyperelliptic surfaces, 126, 228, 244 Hyperelliptic Weierstrass point, 326
Intersection matrix, 148, 253 Jacobian mapping, 248 Jacobi variety, 145, 153
Lattice subgroup 135 Leray covering, 46 Line bundles, complex, 53 -, flat, 132
Manifold, topological, 1 Normal Riemann surface, 125 Order, of branching, 220 -, of cross-sections of line bundles, 56 -, of distributions, 80 -, of holomorphic functions, 6 Partition of unity, 35 Period matrix, of Abelian differentials, 145, 253 -, of lattice subgroups, 140, 142 Picard variety, 136, 146, 153 Point bundle, 114 Presheaf, 16 -, complete, 19 Principal curve (mapping), 241 [also called canonical curve] Projective line, 10 Projective linear group, 174 Pseudogroup property, 4, 164 Refinement of a covering, 28 Refining mapping, 28 Riemann surface, 4 Riemann's equality (inequality), 148 Riemann matrix (pair), 150 Riemann-Roch theorem , III Schwarzian derivative, 167 Sections of a sheaf, 15 Serre duality theorem, 75, 95 Sheaf, 14 constant, 15
fine, 36
-, of germs of differential forms, 68 -, of germs of distributions, 83, 90 -, of germs of divisors, 48 -, of germs of holomorphic functions, 20 Structure, affine, 167 -, complex analytic, 4 -, differentiable, 4 -, projective, 167 -, subordinate, 4, 167 Support, of a distribution, 85 -, of a function, 80 Symplectic group, 155
Torus, 11, 137, 140 Weierstraas, gap sequence, 120 -, point, 123 -, -, normal, 125 -, hyperelliptic, ]2 6 -, -, theorem, 51 -, weight, 192
